Your class III weapons dealer will walk you through the process. It is a bunch of paperwork, prints, etc and a $200 fee and assuming all is clean and in order will take several months to get the tax stamp. Until that time, your dealer will hold the weapon for you.
The problem with full auto in Florida is it is the one thing you can't carry on your concealed permit, so you can't have it with you loaded and ready as a "defensive" weapon making it a range only toy. Plus be certain that you are in full compliance with the Federal transport rules and have the tax paperwork with you even when driving to and from the range as LEO will likely not look real highly on even having a full auto in your vehicle.
Personally, the limits of where and when you can shoot full auto especially here in South Florida to me makes it more logical to simply rent one at the range when I feel the need for some firepower.
Suppressed is another story. A suppressor is actually not disallowed under the Florida CWL so you could have a concealed suppressed weapon available. Again, though, have the tax paperwork available for inspection or expect the weapon to be confiscated until you produce it.