If you are debating which islands to check out, these are ones that I find are usually easier to hunt out meat-free meals. For some Caribbean islands you can ask for or find "Ital" food / cooking (like "Vital" without the V) which is vegan, organic dishes but they may contain honey.
Jamaica
Home to the Rastafarian religion, here you will find tons of vegetarian and vegan options as they do not eat meat.
Puerto Rico
There are quite a few cafe's and restaurants that feature vegan or vegetarian meals and the food chain Pollo Tropical does not use any meat products in their rice and beans. You can also try Asiquesi and Cabo Rojo.
Trinidad & Tobago
This island has a strong immigrant background of Indians and Chinese so you can find a lot of restaurants that cater to those ethnicities which means you can also find a lot of veggie friendly meals. On Trinidad try Mother Nature, Port of Spain and on Tobago try Kariwak Village Restaurant, Crown Point.
British Virgin Islands
Sugar Mill Restaurant at Sugar Mill Hotel is constantly voted the Best Restaurant in the Caribbean by Caribbean Travel & Life. This gourmet restaurant is constantly updating its menu but you can always find multiple vegetarian dishes. Nature's Way on Town Road is a natural supermarket that offers veggie take away options. If you pop over to Nevis then check out Natural Livity and Newcastle Village.