Florida doesnt require in-state ID, I think you can give them a out of state address for the registration but if they require an in-state address, give one (could be a local parking garage if you dont know one). Also give them an out of state one and ask for it to be your "Mailing Address" Florida has a paperwork slot for a seperate mailing address so it doesnt really matter where you register it in Florida as long as you still get the paperwork via mail.
As for can you nuy from Jersey, sure..but depends on the situ.. If its being bought from a private party, you either need to ask them to leave their plates on it (and ship them back once you register.. most people wont do this since you can accumulate parking tickets, commit crimes etc..).. 2nd option is to put an old plate of your own on, you may get pulled over if a cop runs it, but they usually arent to mean if you have insurance on it and a bill of sale..
If you buy it used from a dealer you can just tell them you wish to register it in your own state and ask for "Temp tags".. that will give you 30 days with a paper plate to get your own and pay your state taxes.
If you buy new from Tesla, im not sure.. You should be able to but id call ahead and ask about that process, I believe the order process requests information of where it will be registered so they may already do it for you or again like a dealer they could possibly issue you a 30 day temp tag and let you do it yourself in Florida.
Cheers!