Renting a boat is out of the question. You would need a belize captain's license and that you can't get that without being a land owner. (plus it is dangerous to drive around the island without intimate knowledge of coral hazards)

You can fish off of most docks for small snapper and other small stuff. The very south tip of the island is a pretty good spot if you are up for a bike ride / cart trip.

Or, you can also rent a kayak and fish right on the reef for snapper and barracuda.

To get the big stuff, you will need to hire a boat though.