The question asked in the title of your post doesn't seem to match up with the body of your post. The title is "HELP! ATF Says it is not legal for shooting club to buy guns." That, of course, is patently incorrect, but for a club to buy firearms the club would have to be a corporation.
However, in the body of your post, you seem to be concerned not with whether or not the club can buy firearms, but rather with whether or not members can use firearms without having gone through a background check. And the answer is that of course they can. There is no law of which I am aware requiring a background check before someone can shoot a firearm. While some over-zealous BATFE suit may think that any time you allow a different person to pick up a firearm a "transfer" has taken place, the fact is that commercial ranges rent guns to perfect strangers all the time. I shoot regularly at such a range, and I'm friendly with the owner. I've known him for almost twenty years, and in all the times I've been hanging around the counter at the range shop, I have NEVER seen him call in a background check because someone wanted to rent a handgun.
What he WON'T do is rent to someone whom he suspects of being a prohibited person. A couple of years ago, for example, three "yoots" came in, fully bedecked in jeans that hung down perilously close to their knees, lots of tats, and other hints as to their possible affiliation. They wanted to rent and shoot some Glocks (naturally). As first timers, they were all asked to fill out a form. One of the questions is, "Have you ever been convicted of a felony?"
So one of the yoots got to that question, turned to his companion, and asked (within all our hearing), "Yo! Was dat armed robbery thing I did a felony?" (Not an exact quote, but pretty close.) At that point the proprietor reached over and took the paper away from the guy. He asked if the kid had been in prison, and the kid said yes. So my friend informed him that (a) he wasn't renting any guns, today or any other day; (b) he wasn't shooting any guns his friends rented, today or any other day; (c) if his friends wanted to shoot, he could go down the road for a cup of coffee at McDonalds and come back in an hour when they were done; and (d) that he really needed to understand that if he EVER touched a firearm he was breaking the law and could go back to prison.
The young gentlemen were decidedly unhappy. It appeared they were getting ready to give the owner some serious lip when they looked around and saw maybe four of us regulars, all wearing 1911s on our belts, in addition to the owner and his assistant who were open carrying behind the counter. They ultimately decided that the two who hadn't done hard time would shoot, and their friend would go have a cup of coffee.
So what was my point? Oh, yeah. Background checks are not required. It's only illegal to allow someone to handle or shoot a firearm if you know or have reason to believe they are a prohibited person.