There's no reason to believe that people's information is being automatically sent to the BATF, although on State and local levels, they do what they want to do.
The BATF will handle it like it handles everything else. A pile of tedious (mostly legitimate) record-keeping by licensees, with the occasional audit whenever the poor unfortunate licensee shows up on their radar.
The way that the BATF works it with FFLs, Alcohol vendors, etc., is that they show up in person and dig into your files. There's no automatic transmittal of anything. There's no "put it in the mail" or "email it". Even such tasks such as background checks for firearms, are submitted through a phone call, with the only legal record of the check being in the licensee's files. The information collected for the background check is presumably destroyed at the Federal level immediately after the check is complete. It's illegal for the BATF to maintain such records in the vast majority of those cases.
Not that anyone in DC follows the law.
It will probably be the same for vaping. Record keeping at the licensee level (possibly YOUR city/state as well) and audits, but not any Fed databases scanned by AI at this point in time.
...that's on 17track. USPS tracking still shows "Status not available". 
