He's claiming gun shops are selling firearms without the requisite background checks, which I would wager is a damned lie.
Should be easy enough to prove. Obtain from each manufacturer a list of weapons sold to various gun shops, go to each of those gun shops and confirm that the listed weapons are either (a) still in inventory, or (b) sold, with a valid background check on file. If, for some particular shop, there is a suspiciously large amount of inventory that was lost or destroyed, without a sufficiently reasonable explanation, then maybe that shop is engaged in illegal sales of firearms.
My guess is the inventory flow will show a remarkably high level of confidence upon audit - i.e., very little loss that cannot be chalked up to legitimate causes, such as theft (which the owner was not involved in).