Technically, there's nothing stopping an Oath of Vengeance Paladin from dual wielding swords, so long as you follow the standard rules:

When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative. If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.

So your dual-wielding Paladin would be limited to using shortswords, and wouldn't get to add their Str/Dex mod to the off-hand attack made as part of their bonus action. Overall, it would probably be a bit a step down in terms of damage per round compared to a similar Paladin using a two-handed weapon, but it would give you the opportunity to nova even harder than most Paladins by attacking three time, and smiting on each hit.

You also have the option of taking the Dual Wielder feat for a +1 bonus to AC and the ability to use non-light weapons (say, a longsword) in your main hand.