Yes, if there are spell slots available to cast them with.

The Oath spells add choices, not spell slots, to the Paladin's magical arsenal.

To answer by way of illustration

Paladin, Oath of Vengeance, 5th level, Charisma 16 (+3 mod).

1. How many spell slots does the Paladin have to expend for this adventure day?

   4 1st level spells, 2 2d level spells (p. 83, PHB, Spell Slots per level)

   To cast one of your spells, you must expend a slot of the spell's leverl, or higher. You regain all expended spell slots when you finish a long rest. (p. 84 PHB)

2. How many prepared spells does the Paladin have to choose from when deciding that he needs to cast a spell?

   Preparing and casting Spells (p. 84 PHB)

   You prepare the list of Paladin Spells that are available for you to cast, choosing from the Paladin Spell list. {See PHB p. 208 for 1st and 2d level Paladin Spells}.

   Note that there are more spells available than you can prepare on a given day. How many total spells can you prepare?

   Choose the number of paladin spells equal to your spell casting modifier (+3 for 16 Charisma) plus one half of your Paladin Level, rounded down.

   Before we consider Oath Spells, you can prepare 3 + 5/2 = 5 spells total.

   OK, pick a total of 5 spells from the list on page 208. Let's pick: 3 firsts and 2 seconds for simplicity
   Bless, Command, Cure Wounds (1)
   Magic Weapon, Lesser Restoration (2)

   The Paladin can cast 4 first level spells, and 2 second level spells before running out of slots, and cannot cast any spell that was not prepared. (Such as Wrathful Smite, since that spell was not prepared). The paladin can instead cast 6 first level spells, by using a two second level slots to cast, for example, Cure Wounds cast at second level twice on grievously wounded comrades.

3. How do Oath Spells Figure into this?

   Oath Spells increase the number of choices, not the number of spell slots. Add the following prepared spells for free to your prepared list.
   Bane, Hunter's Mark
   Hold Person, Misty step. (P. 88 PHB)

   The paladin has the same 4 first level slots, and the same 2 second level slots, but can now choose from additional spells that were otherwise not prepared from the list: these are always prepared due to the oath chosen.

   The choices, now, when choosing to cast a spell and having a slot to expend, are:
   Bless, Command, Cure Wounds (1) Bane, Hunter's Mark
   Magic Weapon, Lesser Restoratino (2) Hold Person, Misty step
   Whichever spell is chosen, a spell slot is still expended when the spell is cast.

   When the paladin gets to sixth level, he can prepare six spells (+3 + 6/2 = 6) from the list, and still has the Oath Spells prepared "for free" even though the paladin must always a slot to cast any spell.

Vow of Enmity does not require concentration

Nowhere in the description of a paladin's Channel Divinity (PHB, p. 85) or the Vow of Enmity option (PHB, p. 88) does it mention that it requires concentration, so it doesn't. Using the ability simply activates it, and it does not require concentration to maintain.

Concentration is a rule that usually applies to spells, as detailed on page 203 of the PHB under it's own section. Other features that "hijack" the concentration rule make reference to spells, using wording such as "as if you were concentrating on a spell"¹. In other words, if a feature is meant to include concentration, it explicitly says so.

RAW, using this option would not have affected your concentration on the hunter's mark spell.

¹ As an example of this wording, stealing from this related question, an example of a cleric's Channel Divinity option, specifically the Trickery Domain cleric's Invoke Duplicity ability:

As an action, you create a perfect illusion of yourself that lasts for 1 minute, or until you lose your concentration (as if you were concentrating on a spell).

Hence this example would interfere with concentrating on a spell, but only because it explicitly says that it uses the concentration rule.