When you arrive on the peir, they will give you a form to fill out. The form covers many different health items that determines if the company will consider you a risk.
I signed up a few years ago with my doctors verbal approval. At the pier I was rejected due to a history of lung collapses. They still let me on the trip with my wife so I could watch her. I got my money back and was bummed out but enjoyed watching the others. Unless your weight is a safety issue, you should be able to go.
If you go to your cruise lines shore excursion page it should tell you the limits and restrictions for your shore excursion.
You mentioned size as an issue, my DH is a big man, and he couldn't dve because he couldn't breathe correctly with the regulator, well come to find out after our cruise and talking to a friend who is a SCUBA diver the problem was that his wet suit was too small and it was constricting his chest, so if you do go make sure that you get a wet suit that fits properly, it should be snug but not restrictive
Normally all equipment is supplied...wet suit not necessary since you will not diving deep where water is colder. Regular restrictions are pregnancy, diabetes, epilepsy....talk to your operator beforehand and they will tell, sometimes cruise lines dont tell you all and you arrive and need to cancel which is a loss of time...look for a good operator onilne!