If it's an island vacation, look into scooters to go along with the car. Or at least bikes.
If you're not doing most stuff together, is plenty of other stuff in walking distance? If it's Galveston, I'd say hire a second car, and then ask what the hell are you going to Galveston for anyway? If it's Nantucket, which rhymes with bucket, I'd say it depends on what part of the island you're on, and wonder why you're going there in January?
Other than that, the only Island vacations I recall were on Capri (hire a driver if you need one) and Britain (there was pretty good public transit, and taxis, and people kept driving on the wrong side of the road).
Really, I think it would depend on whether you feel that (1) there's no public transit, (2) there's no private alternative for what little you'll need a second car, (3) there's easy parking, and (4) stuff you want to do is just not walkable.