When you look at the statute (ironically they use the medical marijuana residency statute), the stay must be for at least 31 days. A rental agreement for that length of time would work, although that would require you to be here for about 2 months - 31 days to get the first shot and then another 21-28 days for the second shot.
The law also allows an affidavit from a resident for a person who is staying with someone here and not actually renting a place. For example, my mom is staying with us for several months this year and she is covered by our residency.
Florida is not trying to block out snowbirds - the governor is trying to stop vaccine tourism, both international and nearby states. International is the big one, though. South America (Argentina, primarily) and Canada seem to be the sources. The US vaccine tourism stopped when all of the states went to 65+.
People will still find a way around it, but this is about the best that can be done short of restricting it to people with Florida driver's licenses only.