Most clubs are based at courses; e.g., the Podunk Municipal Men's (or Ladies) Golf Association. But the USGA also allows "clubs without real estate."
That means that you and a group of your buddies (at least 10) can form a club, then form a handicapping committee, and then, with the appropriate guidelines from the USGA in place, begin issuing handicaps.
But your club must be authorized by and affiliated with licensed USGA organizations (such as your state golf association) for the handicaps to be recognized as official.
To search for golf associations in the U.S. that are licensed to use the USGA Handicap System, visit this page on the USGA site.
For an individual wishing to acquire a USGA Handicap Index, the first step is ask the pros at your home course for help. In many cases, they will likely be able to get you started quickly through the club's own resources.