To my knowledge, ATF only registers the first transaction involving new firearms, between dealer and purchaser. After that, any sort of registration that happens is purely voluntary and done by whomever is involved in the transaction.

The NCIC tracks "wanted" firearms by serial number, make, model, and caliber...assuming that someone filled out paperwork to request that the stolen firearm be listed as "stolen" in their database.

What is being requested in the letter is the in-house record of sales of the item(s) described, no doubt banking on voluntary compliance in the process.