The ATF does not have a database of gun owners, with the exception of guns that require a special permit like fully automatic weapons. Such a database would actually be illegal. The only way to determine the identity of a person who buys a gun would be to obtain the records for the dealer who sold the gun, dealers are required by law to keep records of gun sales for at least 20 years after the transaction.
OA says that Grimes' handgun is legally registered to him in his home state of Arkansas. However Arkansas does not require a permit to own a handgun nor do they have a gun registry. In fact most states don't have such things, only six states have laws that require gun registration.
At the initial investigation, Scola mentions the restaurant does not have security cameras. However, as the view pans to Scola, in the very top corner of the blue ceiling, is a Security Camera.