Have you seen photos from the island? Then cameras can work there. The photos you will take will be new and unique, so there is every reason to take a camera with you.

While I have never been to a disaster zone like that one, I have been to the jungle which is one of the harshest places for digital cameras. The main concerns are high humidity, which the Caribbean will certainly have, and power. The solar panel is the way to go and I used it for 3 weeks straight but you need a spare battery. Each day you shoot with one, while the other ones charges since you cannot charge at night.

For humidity, bring plenty of ziplocks and keep you camera in there with silica gel to keep moisture out. Also when going in and out of areas with different temperatures, keep the camera in the bag and let it adapt slowly to avoid condensation.

A submersible case is a possibility but probably overkill. If your camera can fail, it can fail at home too but in the Caribbean you will need to keep shooting, so the best thing to do is bring a backup camera and lens, or it could be a small rugged compact, depending on the quality of photos you want to bring back. Having two identical cameras is more practical since you can reuse lenses, batteries, memory cards and charges, so I usually travel that way and only carry a different camera as third one.