Firearms do not have to have a serial number unless they were made by a licensed manufacturer or imported by a licensed importer after the Gun Control Act of 1968 went into effect.

There are millions of pre-1968 firearms with no serial number like the kind that used to be sold over the counter at Sears or JC Penney or Western Auto and such. Home-built non-NFA firearms do not have to have serial numbers unless state law says otherwise.

Some dealers won't deal with unserialled firearms because they're ignorant or scared and haven't read the back of the 4473 form, which explicitly details how to handle transactions involving unserialled firearms.

Other than finding a non-ignorant dealer to handle the background check if necessary, there's no problem selling such a firearm.