Sneak Attack\

You know how to strike subtly and exploit a foe’s distraction. Once per turn, you can deal an extra 1d6 damage to one creature you hit with an attack roll if you have Advantage on the roll and the attack uses a Finesse or a Ranged weapon. The extra damage’s type is the same as the weapon’s type.

You don’t need Advantage on the attack roll if at least one of your allies is within 5 feet of the target, the ally doesn’t have the Incapacitated condition, and you don’t have Disadvantage on the attack roll.

The extra damage increases as you gain Rogue levels, as shown in the Sneak Attack column of the Rogue Features table.

So Sneak Attack says you can do it once per turn when you hit with an attack roll. Am I correct that this is just any attack roll and not a roll as a result of an Attack action?

e.g.; I attack someone standing next to me with my shortsword and hit them. I roll the damage for the shortsword + the damage for Sneak Attack + my Ability modifier, and I Vex them. They move away but do not Disengage because they want to make a different action. I get an Attack of Opportunity. Because of Vex I have Advantage. If I hit them again do I once again add Sneak Attack since it is the same Round but a different Turn, or am I missing something?

Quote from esampson >>

Sneak Attack\

You know how to strike subtly and exploit a foe’s distraction. Once per turn, you can deal an extra 1d6 damage to one creature you hit with an attack roll if you have Advantage on the roll and the attack uses a Finesse or a Ranged weapon. The extra damage’s type is the same as the weapon’s type.

You don’t need Advantage on the attack roll if at least one of your allies is within 5 feet of the target, the ally doesn’t have the Incapacitated condition, and you don’t have Disadvantage on the attack roll.

The extra damage increases as you gain Rogue levels, as shown in the Sneak Attack column of the Rogue Features table.

So Sneak Attack says you can do it once per turn when you hit with an attack roll. Am I correct that this is just any attack roll and not a roll as a result of an Attack action?

e.g.; I attack someone standing next to me with my shortsword and hit them. I roll the damage for the shortsword + the damage for Sneak Attack + my Ability modifier, and I Vex them. They move away but do not Disengage because they want to make a different action. I get an Attack of Opportunity. Because of Vex I have Advantage. If I hit them again do I once again add Sneak Attack since it is the same Round but a different Turn, or am I missing something?

It's once per turn, not once per round, so you are not missing anything.

This was also answered in the Sage Advice Compendium (EDIT: removed duplicated text):

Can a rogue use Sneak Attack more than once per round?

The Sneak Attack description specifies that you can use the feature once per turn, but it’s not limited to your turn. The feature also doesn’t limit the number of times you can use it in a round.

You sometimes get a chance to use Sneak Attack on someone else’s turn. The most common way for this to happen is when a foe provokes an opportunity attack from you. If the requirements for Sneak Attack are met, your opportunity attack can benefit from that feature. Similarly, a fighter could use Commander’s Strike to grant you an attack on the fighter’s turn, and if the attack qualifies, it can use Sneak Attack. Both of those options rely on the use of your reaction, so you could do only one of them in a round.

Because you get only one reaction per round, you’re unlikely to use Sneak Attack more than twice in a round: once with your action and once with your reaction.