Prior to the Gun Control Act of 1968, people could (and did) buy guns through mail order, with the shipper neither knowing nor caring the age of the buyer as long as the check cleared.
Also prior to that act, you could walk into a store and buy a gun for cash without needing to give ID.
Correct. I purchased my first pistol (Colt Woodsman) at age 17, in a neighboring state, for cash across the counter. No ID required.