We had a cat on board, and never had a problem when clearing into the islands of the Leewards and Windwards. We stated in the declaration form that she stayed on the boat
and would never go ashore. It's much more difficult for a dog to stay on board at all times.
On the islands that are rabies-free, you will have issues if you try to take your dog ashore. You may be able to convince the customs/immigration agent that your vaccination paperwork is enough to allow the dog to come ashore to "do it's business" and get a little exercise, or maybe not. The rules can be very unevenly enforced.
I talked with a cruiser in Bequia who had taken his golden retriever ashore for a potty break. A customs
agent saw him and told him that if he brought the dog ashore again, it would be shot on the spot.
You should try to find out which islands are rabies-free and trying to stay that way, versus those that have rabies and are therefore not restrictive. There are Internet
sites that can help. For example, I recall
is lenient about letting dogs
ashore, but I've also heard of cruisers' dogs
being poisoned by eating poison-laced food
left out by locals there. They do this to protect their sheep and goats from feral dogs.