We love to ride, but I wouldn't want to do it in most of the leewards or windwards. I could see riding on st. kitts and nevis, but most of the other islands would be too hilly, the roads suck, and it's too hot.
As for provisioning
, buy a good folding cart and you'll have no trouble walking to and from the stores. The odd time it may pay to take a bus or cab, but I really don't think you'll miss your bikes.
If you want exercise, there are lots of great hiking opportunities. (and swimming, of course)