Assuming you are an Ohio resident, the first question is whether the other party is also an Ohio resident? If not, then you absolutely must use an FFL.
If the person is an Ohio resident, then it's up to Ohio. Some states have no restrictions on private transfers, other prohibit them entirely, and some states are in between. An example of the latter: In PA, private transfers of long guns are OK, but handguns require the parties to make the transfer at an FFL. A bit of searching indicates Ohio has no restrictions, except that the person on the receiving end may not be a prohibited person:
https://consumer.findlaw.com/consumer-transactions/private-gun-sale-laws-by-state.html
Now whether it's a good idea for you to accept these firearms w/o going though an FFL is whole 'nother question, which I don't feel comfortable trying to answer.
Without an FFL, I'd want a written bill-of-sale listing the makes, models, serial numbers, both parties' contact info, and I'd probably put copies of both parties' IDs on it was well. This will protect you in case one of more of the guns has some kind of illicit history, and it will protect the other party if the authorities ever question what became of the firearms he used to own.
If you do get an FFL involved, I do not think the seller has to undergo a background check. I believe the FFL will wish to record the seller's info and will likely want ID, but I don't think that's required by law or regulation.
Note: I'm giving you this from memory.
The only firearms auction site I'm at all familiar with is
Gunbroker. I have purchased some guns that way years ago; they were shipped to a local FFL, where I paid the fee to have them transferred. The FFLs were never particularly interested in where the guns came from; just that I (as the buyer) could pass the background check.
PS: He needs to get rid of any and all ammo as well. I doubt it happens often, but I remember reading about some case where the authorities were pissed at someone who was a prohibited person and busted him after finding nothing more than a few loose cartridges in various calibers.