There are a number of 'special' vaccination sites in the USA which are indeed vaccinating first come, first served people without going through any special qualifications/eligibility check. For instance FEMA was, and still is, running a mass vaccination site in Yakima, WA which was a massive hot spot of infections in WA state.
They quite specifically told the media that they were focused on getting "shots into arms" and not spending a lot of time on each person checking residency documents.
https://www.google.com/search?q=yakima+wa+vaccination+site+f...
Although by the time it opened it was also possible for most people in the Seattle metro area to get a shot without a very long wait, I do know a few people who drove over there and quite clearly showed their drivers license for ID, and got vaccinated without violating any policy or lying about anything whatsoever.