There is a special area for those in wheel chairs or scooters inthe theater, so on the ship should be no problem. It is very accessible.
However, the ports are going to be another story.
In Labadee, I remember seeing chairs with sand wheels (made to be pushed in the sand, that is) that would open the island up for someone in a chair. A scooter is going to be tough on Laqbadee.
Cozumel would be OK if you disembarked and then took a taxi to Chankanaab for the day (or one of the other accessible beach clubs). Downtown Cozumel would be very diffcult for someone in a scooter. However, if you dock at the downtown pier, the new mall is accessible for shopping.
Cayman is going to be tough. It is a tender port and the tender wharf is not accessible friendly nor is the downtown area, in my opinion.
Ocho Rios will be fine on a scooter.
Hope that helps.