Hi, welcome!
You can rent a car and drive it as far as the border of Belize but you won't be able to take it across, as I understand it. However, that might be for the best because it is easier to get to the diving at Ambergris Caye by fast boat public transit from Coronal, Belize anyway. Once on Ambergris Caye and you are ready to go to Roatan, Tropic Air has flights to that Honduran island out of Belize City. Otherwise, it is a very very very long journey by land and sea. I know it seems like they are so close, it should be easier to do. A number of years ago, I did a back to back trip (one week Roatan, one week Belize) and had to fly back to Houston for an overnight before flying back down to Belize. Yes, there were other ways to take the trip at that time but they were even more convoluted, much less comfortable flights and cost more in time and money.
I'll happily provide more detailed info on the border crossing, the ferry to Ambergris Caye and then the flight to Roatan, if you are still interested in that plan. Let me know.