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The 7th level Vengeance Paladin class ability states
When you hit a creature with an opportunity attack, you can move up to half your speed immediately after the attack and as part of the same reaction.
If you are mounted, is your speed that of the mount or just your own natural speed?
Please answer with regard to independent and controlled mounts as well as Find Steed and Find Greater Steed to make this question as broadly usable as possible.
Being mounted does not change your speed. Being on a mount only means that you move when your mount moves. And while you might control that movement, it is not yours mechanically. That is why they have to specify that when your mount provokes an attack of opportunity, you can also be targeted (PHB 198). If it was your movement, you would obviously provoke the attack.
Whether you can use the movement gained from this feature to dismount is questionable. You can only dismount
Once during your move (PHB 198)
which generally means your movement on your turn. A bit more liberal reading could allow you to dismount as you gain a "move" from the feature. This will be up to the DM.
Thus strict RAW says that you cannot use the feature at all while mounted. As DM, I personally would allow a character to dismount using the movement gained from Relentless Avenger.
the mounted combat rules state:
The initiative of a controlled mount changes to match yours when you mount it. It moves as you direct it, and it has only three action options: Dash, Disengage, and Dodge. A controlled mount can move and act even on the turn that you mount it. An independent mount retains its place in the initiative order.
The mount and you may share initiative if you control it. However, despite possibly sharing initiative scores, you each still have your own distinct turns and statistics. That means that your own statistics, including your speed, do not change when mounted. Effectively, the mount may be carrying you around but it's doing so by using its own speed; you're not using your own. This is true regardless of whether you are personally directing the mount or not.
Since the paladin's class features do not specifically extend to any potential mount, the speed referred to in this feature is the paladin's speed.
Since the Relentless Avenger class feature only applies to you, not to your mount, the mount cannot move using this feature. This would mean that to use it, you would have to dismount and run the distance on foot. The dismounting rules state:
Once during your move, you can mount a creature that is within 5 feet of you or dismount. Doing so costs an amount of movement equal to half your speed.
Assuming that you can dismount using the movement granted from this feature, all of the movement the feature allows would be used up by the dismount.
