Actually it must be pretty nice to have such a "problem" , when we fly if someone shows up flying at our site we encourage it, there's just not enough people out here flying to ever want to run one off.
If you have an organized field and if it has an AMA requirement, then that's another issue and you need to let persons know that 'no card means no flying' and back to getting to what really matters, I've bought a kid an AMA card as a kindness gesture because I knew $ to be tight and he was so happy that the next year he went and proudly bought his own...
No dramatic policies, just saving the world one pilot at a time.