There are campgrounds run by the NPS located throughout the park. Sites are fairly basic, though there are showers and toilets. Depending on the time of year, the cold water can be either bracing or a welcome relief. The NPS information offices provide a map of all campsites, as does the park website.
There are also many backcountry campsites ($2 per night). For these you'll also need a permit ($15) and to reserve ahead (at the visitor center) before disembarking. Sites are free during the off season (May to October). Note that these sites are accessible only by canoe or kayak (excepting one site reachable on foot from Flamingo).