If the dealer has the original records, it may be available, but there is no federal registry. See link ... the ATF is not authorized to keep purchase records to the best of my knowledge. This is because it was a major concern over the years as new gun control laws were enacted. The fear was, and remains, that the feds could pass new laws and then begin to hunt down existing owners of something that was legally purchased. Gun law is all very, very complicated and political.
The ATF certainly does keep a registry of 4473 forms, though their main system for storage is on microfilm, and is beyond inefficient. See: https://www.thetrace.org/2016/08/atf-non-searchable-databases/
If they ever cared enough to run all those miles of film through a scanner, they would have a fully indexed and searchable database of every FFL dealer sale since 1968, complete with names, social security numbers, and serial numbers.
Also, the ATF is now pushing a free software package to FFL holders which allows the 4473 to be filed via computer, meaning its fully indexed and searchable by default.
The privacy law you're thinking of is the privacy act of 1974 which has incredibly broad conditions for disclosure within the federal government itself. Funny thing, this law doesn't actually apply to the ATF under any conditions. The law pertains only to agencies, and the ATF is a bureau.
I always assumed the serial number was included in the mountain of paperwork they have to fill out for gun sales and was entered in the database.