I don't see an issue here. No new laws, no new regulations on guns. Just check that you are legally entitled to own a gun.
You could log into some online DB with your ID/PW and run the name of the gun buyer. Should be quick and easy. Get a print out or .pdf file that they can buy as many guns as they chose to buy.
People who think they know everything are a great annoyance to those of us who do.