An-y-bo-dy.
James Taranto makes an excellent point about the major problem with proposals to expanding who can request a background check on firearms:
Currently access to the FBI’s background check system is limited to licensed firearms dealers, who have an incentive not to abuse it lest they lose their license. If it’s opened up to all prospective sellers of guns–that is, to everybody–what’s to prevent someone from abusing it, say by requesting a background check on Greg Sargent, who presumably has no interest in acquiring a gun?
The system only gives a yes-or-no answer as to whether the putative buyer is eligible to own firearms under federal law. But if you’re looking to dig up dirt on someone, a “no” answer on a firearms background check would give you a nice clump of it.