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.

Note: This answer is for dnd-3.5e exclusively. Other answers may provide pathfinder specific advice.

The Monster Manual on Slaying a Vampire says, "Exposing any vampire to direct sunlight disorients it: It can take only a single move action or attack action and is destroyed utterly in the next round if it cannot escape" (253). Libris Mortis expands on this description, that text on Sunlight Vulnerability saying that this means

…direct exposure to the light of the sun. Reflected sunlight, whether via a mirror or the moon itself, has no effect on an undead creature vulnerable to sunlight. Cloud cover or similar interference does not protect the undead creature unless it is thick enough to provide concealment to the creature. For example, a vampire within a fog cloud spell would not adversely be affected by sunlight. Even thick clothing, as long as it covers the body completely, can protect an undead creature from the dangers of sunlight. (140)

That text's Table 7–3: Undead and Sunlight (ibid.) says that sunlight exposure leaves a vampire "disoriented, destroyed" then provides a note saying, "One round after exposure to sunlight begins, a vampire that remains exposed is destroyed utterly." However, the Monster Manual remains the primary source for a vampire, and that means—despite Libris Mortis—during that 1 round the vampire still exists after it's exposed to sunlight—when the vampire's disoriented—a vampire is still limited to only either a move action or a standard attack. This limitation on the vampire's actions makes it incredibly difficult for a vampire to activate effects that can rescue a vampire from its impending destruction.

For example, even if a DM were to allow the effect to function in such a way—and most won't, making that an incredibly big if—, blotting out that pesky sun via the effect of the Iron Heart maneuver iron heart surge [special] (Tome of Battle 68) is impossible for the typical disoriented vampire as such a vampire must take a standard action to employ the maneuver. Further, for example,—assuming the rules for doing so from the Rules Compendium are in use (85)—activating a wand of quickened no light [trans] (Book of Vile Darkness 100) (0-level spell at caster level 1 modified to a 4th-level slot by a level 7 caster) (420 gp/charge) is also impossible, a disoriented vampire being unable to take even free actions, much less a swift or immediate action.

With all this in mind, for a vampire on a budget, listed below are what I think are the most cost effective ways for a dnd-3.5e vampire to avoid destruction from exposure to direct sunlight.

Wear clothing that "covers the body completely"…

If the DM's hewing closely to the rules, an explorer's outfit (PH 129, 131) (10 gp; 8 lbs.) probably won't be enough to cover the vampire's body completely, so a cold weather outfit (PH 129, 131) (8 gp; 7 lbs.) likely won't be enough either. A vampire in such a campaign needs an outfit that says it covers the wearer from head to toe, face included. The black bodysuit (Arms and Equipment Guide 29, 30) (30 gp; 1 lb.) may be sufficient, although to reap its other benefits also the vampire must be lightly equipped. However, the most foolproof method of covering everything according to the rules is probably the hydration suit (Sandstorm 99, 101) (1,000 gp; 10 lbs.), essentially a Dune-style stillsuit with goggles and everything (from the novels not the films). However, a more generous DM could rule an explorer's outfit sufficient, maybe supplemented with piecemeal clothing from the Arms and Equipment Guide's Table 2–2: Clothing (29).

However, for a vampire, the terrifying thing about relying on this method to protect it from a sunlit environment is that clothing—like other worn items besides armor—is vulnerable to sunder attempts… and sundering a worn object is particularly easy: not only is doing so usually easier than hitting the vampire, but also clothing typically doesn't have much in the way of hardness or hp. Seriously, once a dnd-3.5e vampire that's out in the open on a sunny day is recognized as a vampire, the good folks of the town should do their best to rip its clothes off.

…Then take the feat Endure Sunlight…

The monstrous feat Endure Sunlight (LM 26) has as its prerequisite a vulnerability or weakness to sunlight—like that of the typical vampire—and has as the following benefit :

You can resist all dangerous effects of sunlight for a number of rounds equal to 1 + your Charisma modifier (minimum 1 round). After this time, if you are still exposed to sunlight, you take the normal effects as appropriate for your kind.

This gives any vampire at least 1 round to do something other than move or attack, like activate a wand of obscuring mist [conj] (PH 258) (1st-level spell at caster level 1) (15 gp/charge) (just as effective as the 2nd-level Sor/Wiz spell fog cloud [conj] (PH 232) for the purpose of saving a vampire from destruction), that aforementioned wand of quickened no light (without the quickening works, too!), or even—if the DM says it'll make a difference—the martial maneuver iron heart surge. However, if the vampire wants true reliability in avoiding its destruction, it should probably have on hand a very particular potion (see below).

…Then chug potions of cloak of dark power

A potion of cloak of dark power [abjur] (Spell Compendium 48) (1st-level spell at caster level 1) (50 gp; 0.1 lbs.) should be available in all but the smallest town, despite the spell itself being available only as the 1st-level spell of the domain Drow (273). The spell's effect creates around the subject a dusky haze that "does not interfere with vision, but the subject and anything it wears or carries is protected from the effects of full sunlight, even under the open, daytime sky of the surface world." The spell's 1 min./level duration should be sufficient for a vampire to get to safety after the vampire's clothes have been ripped off and its rounds of sunlight endurance nearly exhausted.

If the DM rules the potion isn't available—a strong possibility given the rarity of casters who can trigger the spell so as to create a potion of it—, a vampire can take a full-round action (that somehow doesn't provoke attacks of opportunity) to spread on itself the alchemical substance liquid night (LM 74) (150 gp; 0.5 lbs.). Doing so protects the vampire from sunlight for 1 hour. The cost—both in actions and gp—is higher than for the potion, but a vampire lacking access to the domain Drow may be able to itself manufacture liquid night instead of needing to purchase potions of cloak of dark power from increasingly suspicious vendors. However, both the potion and liquid night have achingly obvious effects—the haze of the potion's effect and the coating of liquid night being a "dark, sticky fluid" that "has a distinct musky odor of moonflower"—, making the presence of either effect a (ahem) dead giveaway that something's not right with the affected creature.