Answer: You can grow any vegetable you like! The planting seasons are Sept./October or Feb/March. Not much will grow during the summer months so plan your garden around June, July and August. Here's a publication from University of Florida that you might find helpful: http://edis.ifas.ufl.edu/VH021
I don't know what plants are affected by black spots so I'm not sure what to suggest. Black spots can be either fungal or bacterial in nature and not all control recommendations are safe for all plants. If you submit another question with specific plant names we may be able to help.
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