If you mail order, the FFL that does the transfer is responsible for verification of your eligibility.
From the wording of the bill, all FFLs will have access to the database. So Cabela's in Nebraska will be able to see if you have your permission slip. Will this happen? Probably, at least among the big vendors. They just add another line in the computer.
This is a very clever bill, no doubt.
Watch for it to be introduced in every state over the next few years- at least until the Supreme Court rules on it's constitutionality.