Most of the cenote tours are run on the mainland. To get there from Cozumel, you take a ferry (about half an hour) and then you need transportation from Playa del Carmen to the caves (about 30 to 60 minutes, depending on site). At least some of the Cozumel dive ops can arrange a day of cenote tours for you, or you can arrange it with a mainland dive op who will meet you in PDC with a van.
There are caves on Cozumel, and German Yanez is a cave instructor/tour leader who lives there, but I don't know if he leads cavern tours in the Cozumel caves. At least in the one I've dived in (Aerolito) there wasn't that much of a cavern zone.