Depends on your local laws. I know in Mississippi you could, as long as you don't deal in NFA or any other restricted/illegal items. It's called a private sale/purchase.
As long as the firearms are legal to own in you state, I think you could. Though, you would probably get mostly Saturday Night Specials out of it. I'd also have each gun checked out to make sure it was not stolen. You'd probably be better off going around to flea markets, and pawn shops looking for a good deal.
Check local laws for any problems you might run into. Are you really prepared to give out $50 in cash for rusty pieces of garbage? The only reason municipal governments can afford to to buy-backs is because they're not using their own money. They're using ours. (And before you ask, no, you can't levy a tax first to fund your buy-back. )
I would be worried about inadvertently obtaining stolen weapons or having people try to get rid of firearms with serial numbers removed... I frequently find ads from people offering to buy broken and/or old firearms on sites for the purpose of private firearms sales.
Schuss- REALLY not a good idea. Say that the gun is illegal due to alterations (cut down barrel below legal limit, defaced serial number, etc) You REALLY do not want to be found holding one of those pending a govt buyback. The felony conviction that will follow will keep you from possessing ANY firearm.
