update
In case someone is interested:
After some research and having contacted a few ranges in FL, that caters towards tourist, I have found out that it is indeed illegal for a non-resident alien to possess fire arms and amunition in the US. There is however a few exceptions and the one used by ranges in large seem to be that you are exempt if you possess a valid hunting license.
So these ranges sell you a hunting license with your range time.
In case someone is interested:
After some research and having contacted a few ranges in FL, that caters towards tourist, I have found out that it is indeed illegal for a non-resident alien to possess fire arms and amunition in the US. There is however a few exceptions and the one used by ranges in large seem to be that you are exempt if you possess a valid hunting license.
So these ranges sell you a hunting license with your range time.