Yes (probably)

Rules as Written, I think you can kill the 0-hit-point Misty Vampire by doing damage to it three times before it escapes to its resting place, or by doing massive damage to it (equal to or exceeding its maximum hit points).

But there is certainly room for disagreement, as the rules regarding creatures with 0 hit points are written in the context of creatures that are unconscious at 0 hit points, which the Misty form of the Vampire is not.

Honestly, I don't think the intent of the rules is to allow a Vampire to be destroyed in this way. It seems that the intent is that you have to follow it back to its resting place, drive a stake through its heart, and then destroy its body while it's paralyzed.

Rules around having Zero Hit Points

PHB p. 197 says

When you drop to 0 hit points, you either die outright or become unconscious....

Creatures that take massive damage die outright.

Otherwise they fall unconscious but are dying -- they must make death saves to determine if they eventually stabilize or die outright.

There are three exceptions wherein a creature may be stable at 0 hit points -- still unconscious, but not needing to make death throws.

1. If the attacker decided to strike an incapacitating blow rather than a fatal one to take the creature down to 0 hit points [PHB p. 198]
2. If the creature is stabilized by another creature (by making a medicine check [PHB p. 197], using a Healer's kit [PHB p. 151], or by using the Spare the Dying cantrip)
3. If the creature stabilizes by rolling three successful death saving throws before rolling three failed ones.

Note that in all cases here, a creature with 0 hit points is unconscious.

Normally there is no way for a creature to be at 0 hit points and still be conscious and taking actions. However, the Vampire is clearly an exception, based on the text of its Misty Escape feature. It is very unclear whether any of the usual rules that apply to unconscious creatures at 0 hit points also apply to Misty Vampires at 0 hit points.

So What Rules Apply to a Misty Vampire that is Stable and Conscious at Zero Hit Points?

The description of the Vampire clearly admits the possibility of doing damage to the Misty Form by magical means.

Taking Damage at 0 Hit Points (PHB p. 197):

If you take any damage while you have 0 hit points, you suffer a death saving throw failure. If the damage equals or exceeds your hit point maximum, you suffer instant death.

And:

On your third [death saving throw] failure you die.

Note that a creature that takes damage while at 0 hit points that isn't enough to kill it outright remains at 0 hit points. 5e does not have a concept of negative hit points.

Two Ways to Kill the Zero-HP Vampire (RAW):

1) damage it three times, causing it to fail three death saving throws before it has a chance to stabilize or be stabilized by any of the means listed above

2) damage it once, for an amount that equals or exceeds its maximum hit points (perhaps a massive Paladin critical smite)

Caveats

1. If the Vampire has not used up its Legendary Resistances for the day, it can choose to succeed on its first three death saving throws, so you would have to hit it additional times to get three failed death saves that "stick".
2. It is very unclear whether any of the usual rules that apply to unconscious creatures taking damage at 0 hit points are also meant to apply to conscious Vampires in Misty form at 0 hit points (but RAW I believe there is nothing to override them).
3. It is possible that the Vampire's Misty Escape kicks back in and stabilizes the Vampire every time it gets hit, removing method 1 as a possibility (but I don't think so, see below).
4. It is possible that Vampires are meant to be immune to failed death saving throws (but I don't think so, see below).

On Misty Escape Re-stabilizing the Vampire

Clearly when the Vampire first drops to 0 hit points, the Misty Escape feature permits it to be stable and conscious where ordinarily it would be neither of those things.

If the Misty Escape feature triggers each time the Vampire takes damage while at 0 hit points, it could be argued that the feature stabilizes the Vampire each time.

However, the trigger for Misty Escape is that the Vampire drops to 0 hit points (outside its resting place). Getting hit while already at 0 hit points and remaining at 0 hit points does not constitute dropping to 0 hit points.

On Immunity to Failed Death Saving Throws

PHB. p. 198 says that Monsters can sometimes make Death Saving Throws:

Monsters and Death:

Most DMs have a monster die the instant it drops to 0 hit points, rather than having it fall unconscious and make death saving throws.

Mighty villains and special nonplayer characters are common exceptions; the DM may have them fall unconscious and follow the same rules as player characters.

Note that this is given as an option for a DM to make a creature more powerful than a normal monster that just drops dead at 0 hit points. But we already know that a Vampire is not a normal monster that just drops dead at 0 hit points. Its Misty Escape feature is a stronger feature than the standard Death Saves feature that prevents PCs from just dropping dead at 0 hit points. So this clause in the PHB doesn't really apply in this case.

However, I don't see anything here to suggest, whether the DM usually gives monsters death saves or not, that any monster should be immune to dying after suffering three failed death saves at 0 hit points.

I'll dissect this homebrew race on a per-trait basis, and then offer an overall evaluation at the end:

Neutral traits

These are the traits I find are neither over or underpowered:

Ability Score Increases. Your Strength score increases by 2, and your Dexterity score increases by 1.

Pretty standard, nothing under or overpowered here.

Age. Lycanthropes mature at the same rate as Humans, and live up to around 150 years. Alignment. A Lycanthrope is typically of a Chaotic Alignment.
Size. Lycanthropes stand 6 to 7 feet tall just as a Human. Your size is Medium.

All fluff.

Speed. Your base walking speed is 30 feet in your humanoid and hybrid forms, and 50 feet in wolf form.

This is better than what most races get, but it's situational enough to not be anything of note.

Darkvision. You can see in dim light within 60 feet as if it were bright light, and in darkness as if it were dim light. You can't discern color in darkness, only shades of gray and red.

Pretty standard. I like the addition of the red color.

Tracking. You can detect opponents within 15 feet by sense of smell. If the opponent is upwind, the range increases to 40 feet; if downwind, it drops to 10 feet. Strong scents can be detected at twice the ranges noted above. Overpowering scents can be detected at triple normal range. When a creature detects a scent, the exact location of the source is not revealed—only its presence somewhere within range.

This is a cool trait, but I'm doubtful as to its actual usefulness in-game. The range is so short that you're still going to get surprised, and tracking whether you're upwind or if the creature is smelly sounds like it will quickly get tedious. Consider giving the Werewolf Keen Smell or Keen Senses (Proficiency in Perception), like an Elf.

Shapeshifter. You can take 1 action to change into your Hybrid or Wolf Form. While in Hybrid Form you can not wield weapons, or wear armor. While in wolf form you cannot wear armour or wield weapons. You are able to wear certain equipment at the DM's discretion, such as circlets and robes. You can choose to have equipment you are wearing to either merge into your form or be dropped to the ground. You gain no benefit from equipment that is merged into your form.

The ability to change, itself, is pretty standard, however, the restriction on weapons and armor are weird (They can wield Foci and Holy Symbols, but not weapons?) This trait is probably the biggest counter-balance to all the positive traits because of the limitation on what you can hold in your hand.

Languages. You know Common, Wolf and one extra language. You can only speak in Human and Hybrid forms.

Pretty Standard, though I wonder why you can't speak Wolf in Wolf form.

Wolf Empathy. You can communicate with wolves and dire wolves.

Exceptional but not overpowered. Gnomes can speak with small animals, which I've seen can be useful at times, this trait is going to be a lot more useful than its gnomish counterpart.

---

Positive Traits

These are the traits that your Werewolf race gains as benefits, which I rate as over or underpowered based on comparison with existing traits from other published races:

Regeneration. You have resistance to damage from nonmagical weapons that aren't silvered. At the start of each of your turns, you regain hit points equal to 1d4 + your Constitution modifier if you have no more than half your hit points left, and have at least 1 hit point.

Overpowered. This trait effectively gives the Werewolf race 1.5x to double HP, with the resistance alone. Similar races with resistance such as the Dragonborn (1 element), Dwarf (poison) and the Aasimar (radiant, necrotic), grant only 1 or 2 types of resistances and only for situational damage types- a resistance to the three most common damage types is too strong.

The actual regeneration trait is also very strong. The only race I found with regeneration is the UA-Gothic Heroes Revenant subrace, and the Revenant only regenerates 1 HP per turn, your homebrew has the potential to regenerate 5 HP, with a +3 Con, at 1st-level.

Natural Weapons. While in your hybrid or wolf forms, you have a 1D6 Bite and 1D8 Claw attack. Increase damage at level 5 to Bite 1D8 and Claw 1D10. On a successful melee attack you may use your bonus action to make a claw attack on the same target.

This is tricksy. The d6 Bite is ok, but the claw attack basically grants you access to a non-light one-hand Martial weapon you can use to attack with a Bonus Action, normally you'd need Dual Wielder to pull that off. Taken in conjunction with the Shapeshifter trait where you can't wield anything in your hands when you transform, it sounds ok, but then there's gonna be the powergamer who will want to play a Monk in order to get the early d10 unarmed strikes, with flurry of blows.

Then again, if this race will be paired with any class that isn't a monk, he's gonna have no access to magical weaponry, and be forever at the mercy of creatures with resistance to Bludgeoning, Piercing and Slashing damage.

Trip Attack. While in Wolf or Hybrid form you can attempt to trip the opponent as a bonus action after a bite attack. The opponent must make a Strength saving throw equal to 10+ your Strength modifier + your proficiency bonus.

Flavorful and only slightly overpowered- most save DCs are computed as 8 + modifier + proficiency.

Skills. You are Proficient in Survival, Nature, Intimidation and Perception rolls. You have disadvantage on Persuasion checks.

Overpowered. A Half Elf with Skill Versatility only gets 2 skills, this trait grants four. The Disadvantage with Persuasion doesn't counter balance this enough (nor does it make much sense, do you still get Disadvantage in Human Form?).

Saving Throws. You are proficient with Strength and Dexterity saving throws.

Overpowered. No race grants Proficiency in saving throws. These are strictly benefits from picking a class.

Tough Hide. While you are not wearing any Armour or shields your AC becomes 10+ Strength Modifier + Proficiency Bonus (All Forms)

Overpowered. Unarmored Defenses usually don't include Proficiency Bonus as a component. This is because Proficiency is "free" as the character levels up, and it will quickly get out of hand at higher levels. Consider that at 17th-level, and at +5 strength, this character will have 22 AC, even before items that boost strength like a Belt of Giant Strength.

---

Negative Traits

These traits would be the counter-balancing factors for the Werewolf race. I will say, in advance, that these traits are not effective counter-balancers (it's not as discouraging as, say, Sunlight Sensitivity) for all the Positive Traits above. I'll explain why below each one.

Silver Weakness. You are vulnerable to damage from silvered weapons.

Vulnerability is normally really punishing, but in this very specific case, when the vulnerability comes into play, it will feel like you're specifically targeting the PC, As Erik puts so well in another answer.

Also consider that not even actual Lycanthropes in the MM are vulnerable to silvered weapons, and no official race gives you vulnerability to anything.

Full Moon Rage. At the start of a night with a Full Moon you must make a DC13 Wisdom save or lose control of yourself. During this time the DM has Control over what your character does, however every hour you may reroll your saving throw to take back control. Once you make the correct saving throw you retain control for the rest of the night.

Not a huge counter-balance, it will only come up once a month, and the DC is set pretty low (with a chance to break it every hour). What's more, smart players will make preparations if they know the full moon is coming up. They're going to tie themselves to trees or manacle themselves to a cell, which negates what this is supposed to accomplish.

New Moon. During a new moon, you cannot transform, and are damaged normally (without resistances and weaknesses).

As above, it's not a huge counter-balance. Smart players will avoid adventuring at this time of weakness. If you force them to adventure in this scenario, it will feel antagonistic.

---

Overall, this race is overpowered, almost broken. What's really glaringly overpowered are the Regeneration, Saving Throws, Tough Hide, Skills, and how the Natural Weapons scale so fast. The Werewolf race will outshine just about anything at low-to-mid-levels, and be powerless against higher-level opponents with resistances.

The race has too many things going for it, if you take a look at other races, they usually have 3 or 4 defining traits (not including ability score bonuses, darkvision, languages, etc), this race has a whopping 8.

Using this race power-level analyzer, from reddit's /u/aranim and /u/JamesMusicus, this Werewolf scores:

• Ability Score Increase +3
• Silver Weakness -1
• Regeneration +2 (from regeneration) +1.5 (from resistances) = +3.5
• Natural Weapons +1
• Tracking +0.5
• Darkvision +0.5
• Shapeshifter +0
• Trip Attack +0.5
• Wolf Empathy +0.5
• Skills +2
• Saving Throws +2
• Full Moon/New Moon weakness -2
• Tough Hide +2 (being conservative)
• Language +0.5

Total 13 points, nearly triple the score of the poor Dragonborn (4.5), and eclipsing the highest-rated Mountain Dwarf (8).