Of course the correct answer for the question asked is to simply check in with the Contest Director (CD) and apply for "Instant" AMA membership. The CD's sanction package contains Membership applications, along with 30 day receipts.
At any AMA Sanctioned Event, a non-member CAN apply for and become an AMA member.

The above is the answer.

Below are some hints / suggestions.

1. If you apply at an event, do record the CD's name, address, the AMA Sanction # and date. (If not displayed as it should be, then ask the CD.) There are examples where the CD has been somewhat negligent with the After-Event-Paperwork and if AMA fails to respond within about 3-4 weeks, then you know where the trail leads.

2. If the CD doesn't appear to be aware of his ability to perform your request, then simply ask him to OPEN his Sanction Package. OTOH don't ask him while he is in the performance of some important duty such as getting the registration opened, or doing the pilot briefing, etc.

Reading threads here concerning FAA, etc., and what all AMA should be doing "....for me..." are interesting. However I ask you to consider one thing: Only in numbers represented can AMA really obtain consideration, especially with government. If your only concern is your own direct handout, then you are contributing to the override of government and not for consideration of the whole as applies to model aviation.
AMA is NOT a lobby unit, and cannot be such. AMA can make information known reference the well-being of its members. Only YOU, as a member, can lend assistance to that representation. The less government we are subject to is by far the better and that is a direct handout to YOU.
Of course then as an AMA member you have the opportunity to elect AMA Officers that are truly concerned with the better future of AMA and the membership. So there is another hand extended for you to help yourself.

The real benefits of AMA membership are mostly impalpable