As others have said, if the gunshow dealer is not a local guy, he will likely have an arrangement with one or more local dealers. It's in their best interest since folks won't buy from them if it will be too much hassle to get it.
For example, I bought a shotgun from a dealer at the Pinellas gunshow, but the dealer was based out of Orlando. I conducted the sale with the Orlando dealer taking care of money, Form 4473 and the call-in. He then arranged for a Pinellas dealer to handle the 3-day wait. So three days later I go to the Pinellas dealer, he checks my ID, hands me the shotgun and I'm on my way.
On the 3-5 day wait, FL state law preempts all local laws/ordinances with the exception that the state gave counties some leeway on waiting periods. For example in my current county of Pasco, there is no wait for rifles, but in Pinellas and Hillsborough there is a 3-day wait unless you have a CCW permit. I didn't have my CCW at the time I bought the shotgun so I had to do the two-dealer-tango thing.
LL