Can Opportunity Attacks have Advantage? For instance, if a character with the Mounted Combatant feat was to perform an attack of opportunity on a smaller creature while mounted, would that attack have advantage?
Unless the source of your advantage is dependent on something that doesn't apply to attacks of opportunity (like taking the "Attack Action"), you can absolutely get advantage on an attack of opportunity.
In the case of your specific example, the Mounted Combatant Feat states that (PHB, p. 168):
You have advantage on melee attack rolls against any unmounted creature that is smaller than your mount.
There's nothing in that description that explicitly rules out an opportunity attack. As long as your opportunity attack fulfilled all the above criteria, it would get advantage.