Permanent advantage on opportunity attacks is probably too much.

I've always loved the Oath of Vengeance, but hated its 7th level feature. I am going to allow my player to spend their reaction to move half their movement without provoking opportunity attacks, and *then* make an opportunity attack with the same reaction. Swapping the order of the feature keeps with the theme and power of the ability and also accomplishes two things

1. With no gateway to the movement, they can choose to pursue fleeing enemies *after* the enemy moves. They can also easily spend their reaction to move during an enemy's movement, get in the path and activate another opportunity attack.

2. By not requiring the opportunity attack, the free movement allows them to play support as well, another theme of the Paladin class, and something the 7th level feature of other Oaths does with auras and helpful abilities. Movement to protect an ally, body block for them, pass off potions,  be available for lay on hands on their turn, whatever your paladin can think of.

We're going to test this implementation in our campaign. 

Hmm... I've always thought the way it worked was that the creature finished its move before the paladin got their extra movement.

I'm looking at the relevant rules here...

Reactions

Certain special abilities, spells, and situations allow you to take a special action called a reaction. A reaction is an instant response to a trigger of some kind, which can occur on your turn or on someone else's. The opportunity attack, described later in this section, is the most common type of reaction.

When you take a reaction, you can't take another one until the start of your next turn. If the reaction interrupts another creature's turn, that creature can continue its turn right after the reaction.

Movement and Position

In combat, characters and monsters are in constant motion, often using movement and position to gain the upper hand.

On your turn, you can move a distance up to your speed. You can use as much or as little of your speed as you like on your turn, following the rules here.

Your movement can include jumping, climbing, and swimming. These different modes of movement can be combined with walking, or they can constitute your entire move. However you're moving, you deduct the distance of each part of your move from your speed until it is used up or until you are done moving.

Breaking Up Your Move

You can break up your movement on your turn, using some of your speed before and after your action. For example, if you have a speed of 30 feet, you can move 10 feet, take your action, and then move 20 feet.

Moving Between Attacks
...
Using Different Speeds

Opportunity Attacks

In a fight, everyone is constantly watching for a chance to strike an enemy who is fleeing or passing by. Such a strike is called an opportunity attack.

You can make an opportunity attack when a hostile creature that you can see moves out of your reach. To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature. The attack occurs right before the creature leaves your reach.

According to my interpretation of RAW, the paladin's movement follows the reaction which occurs during the enemy's move. Nothing breaks a move except for an action on the part of the moving creature, individual attacks, or changing movement speeds.  So when a creatures starts moving they are committed to moving the full distance they choose, unless they take an action, make an attack, or change movement speeds.

So, if an enemy with a single attack and single movement speed moves out of the paladin's reach, the paladin moves before the enemy completes their move, they have to break their move with an action in order to not complete their originally intended move. The exception would be if the movement of the paladin directly prevented the completion of the move.

I don't like my interpretation. It doesn't seem likely to match with RAI.

Not every separate ability a class or subclass gets is going to be equally strong or useful. Overall, the Oath of Vengeance is certainly strong - arguably one of the best Oaths there are. It doesn't need more buffing. I get that Relentless Avenger is a little wonky, but it's not useless and the rest of the Oath abilities easily make up for it. If you want to change it to be less wonky, sure, but I'd caution against making it more powerful.

It's a really good, if occassional, ability that doesn't really need fixing. Remember that you will get to move in your turn as well so the enemy will not be able to escape unless they use they action to dash. And since you can just use you action to dash to get in close combat with them you will get to use your AoO when they try to leave the next turn and then you just keep following them. Sooner or later you wil wear them down. :)

I agree it's fine. A vengeance paladin isn't supposed to be locking things down, it's supposed to be pursuing them to the ends of the earth. If you really want advantage on those OAs, you can hit the target with your Vow of Enmity.

Is this feature situational? Yeah. It's pretty normal to have one or two of those when your other features are really good in almost any situation.

I don’t disagree that it is weaker than some of the auras. I think thematically that is in keeping with the vengeance theme. However, consider this ability with pole arm master and it become very strong. Enemy moves to you, you can hit them and move away before they attack. With Sentinel even stronger.