I have lived in Miami all my life and always enjoyed backpacking and camping. To the best of my knowledge, despite the enormous amount of wilderness in the area, there is only one outfitter store in the metropolitan area: Jet’s Outdoors. It’s in west Miami, you’ll need a rental car, a really good feel for weirdo public transit systems, or a lot of money for a cab. (http://www.jetsfloridaoutdoors.com/)
There are some other canoe and kayak shops (notably Florida Bay Outfitters, on the way down to the Keys-http://www.kayakfloridakeys.com/), but as South Florida is so hot, flat, and waterlogged, people don’t go backpacking much. Although there is some beautiful country to walk through in the Big Cypress Swamp (not this time of year, though. Only in the winter).
Enjoy your trip!