The law will differ from state to state, but here in Texas, even though it's an auction, you are still purchasing the gun from an individual and no permit or waiting period is required. it's kinda like buying at a yard sale, just pay and go. I've been told that OK. now permits a Texan with a valid concealed permit to purchase firearms there with no waiting period.
If you can't get a straight answer from the local SO, go to your state police, they should be able to guide you in the right direction.
By the way, In my opinion you are looking at a good source for quality firearms at a decent price. I try to make all the yard sales and auctions that advertise guns. I have made some great buys over the years. But by the same token, I have seen guns sell for three-four times what they were actually worth. I recommend you buy a good gun value book, study it, take it with you and be careful when grading the guns, don't allow yourself to get caught up in the biding and pay more than a gun is worth.
Good luck.......I hope you can fill your safe at auctions/yard sales.