Grand Cayman will be a tender port and as such, my experience would be to wait for open tender and then let the crowds die down before going ashore. I regularly push a lady in a chair and the crowds surrounding the tender gangway can be very uncomfortable for someone in a chair. By waiting until there are no lines, you will have a wonderful tender experience. There will be staff on hand to help him down the stairs and onto the tender and also off the tender at the tender wharf in Grand Cayman. Grand Cayman itself is reasonably chair friendly.
Here is a tender trick. Wait until the tender is just about full before you board. You can tell the operator that you want to be the last on and they will generally accommodate you. As one of the last to board, you can grab one of the "pull down" seats and be right next to the stairs when the tender starts unloading. You will be the last to board and the first off. This really relieves the stress for those in chairs.
Cozumel is going to be another story. Depending on which pier you dock at, you will be able to access the shopping areas at the foot of the piers, but downtown Cozumel is realitively unfriendly to those in chairs. It is hot, crowded and rough going. I would suggest simply shopping at the pier shopping areas and then taking a taxi to Chankanaab National Park. There is tons to do there and it is all accessible.
Have a great cruise. Be sure to write a review for CruiseReviews.com about your experiences when you get back. I will look forward to reading it and learning any tips you can share.