Clearly there are hundreds of ranges across the U.S. that allow customers to rent and use guns without first undergoing a background check. That said, those guns are to be used only on the premises and under the supervision of the owner of the guns. Actually, the guns probably are the property of an FFL holder whose designees (employees) supervise the use of the guns.
...buy rifles and shotguns for club use.
The problem is that there must be some entity who actually owns the firearms. If the club doesn't have an FFL, then some flesh & blood person(s) will have to fill out the 4473 and undergo the background check to acquire the firearms.
Can you explain how the club was planning to handle the purchase & use by club members?
1. Who would own the guns?
2. Where would the guns be stored?
3. How would the guns be checked out?
4. Where would the guns be used by those who check them out?
5. Would the legal owners of the guns supervise their use?
Here's one legal way it could be done, in my opinion.
Club official purchases firearm via the standard process and stores it at his home.
Club member desires to use firearm and goes to club official. Club official accompanies firearm and club member to range where club member uses firearm under the supervision of club official. No transfer takes place so there's no issue.