Tampa is rough. St Pete also rough. They do not like anyone they perceive as "homeless" and have local ordinances against "sleeping" and "camping". The Walmarts etc are all "no-go".
I did a few weeks in St Pete, but it was not easy. Along 4th St N, between around 54th and 9th Avenues, there are lots of businesses. I overnighted on the side streets in front of closed businesses. Do NOT park in front of any houses, and do NOT let any neighbors see you--it's a sure call to the cops.
There's a Cracker Barrel in Oldsmar that's OK for a couple nights. I never tried any of the other Cracker Barrels in Tampa or St Pete, but Cracker Barrel policy is to allow campers to overnight.
Good luck.