Cayman, yes. You have to google for Honeysuckle Riding Stables; they don't offer it through Carnival. "Wrong kind of stirrups" or something. They'll take you along the beach, and let you go in the water for a little bit if you like. Three years ago it was like $80/pp. Maybe $90. Pay half for deposit, other half in cash when you get done.
The Mayan ruins horseback riding offered through Carnival on Cozumel is "Mayan" "Ruins". Cast concrete replica statuary. No beach. But they have a long hunting dance in the middle of the ride down in a little cave. If you like that kinda thing [and we didn't] then it's okay. Mexican buffet at the end.
There may be other stables on Cozumel that do beach riding, but I'm not specifically aware of them. You might google for them as well.
Desperately seeking 10th cruise for the free laundry.