Take a look at the rules, "Section 3. Making Application. A licensed or member club that meets all the requirements of the American Kennel Club.
From here go to the AKC website https://www.akc.org/clubs/ and go to Clubs & Relationships then Club Relations then "How to form a club" which will prescribed the process that will qualify a club for the above requirement "licensed or member club". You should get all the answers you need on how to form a club and hold a test. It will take a year.
Now, there is nothing to say a Club can not outsource a test's activities to a company (they are called Event Planners) but the Club has so much responsibility to the AKC in sponsoring a test, I do not see it happening.
Do some research and you will understand that a Club is an organization required to perform certain duties and the AKC is not going to encourage this to become some type of for profit deal.