Police departments do it all the time.?
That's a "sort of."
There's a database of stolen arms with a certain amount of accuracy and completeness.
ATFE has a very incomplete database of their own "traces."
A "trace" usually involves asking the maker of the arm for their records on the s/n. Which is usually "Sold to [distributor/wholesaler]." Then, that party is interrogated, getting an answer of "Sold to [vendor]." Who then gets a call. If that outfit still exists, then they are asked to check their 4473 records book. (Which likely only goes back ten years.) If the 4473 is found, the buyer's information is gleaned. That worthy gets a phone call "Do you own [firearm]?" Which then can fall down a rabbit hole.
Now, if [dealer] is not in business, ATFE calls over to its storage facility for closed dealer Bound Books. There the 13-14 employees can, sometimes, find the correct pile of books, and those books will be legible enough to read. If found, they get the 4473, then it's back to calling the person indicated.
Note the number of ways this trail will go cold.
Like the original firearm was sold to ABC Distributing, who then sold to Big Box Sporting Goods, Inc. who then may or may not have kept records on which of a thousand stores the arm went to. Said chain might not exist anymore, or various stores might have gone out of business--dead ends.
The sale might be 20 years old (that's only 2000). So, the Bound Book will not go back that far. Or, the arm was sold, traded, lost in a poker game, house burned down, owner passed away, etc.
It's a lot easier on tv--but most things are.