If you apply for a JAL Club A card, and get approved you will get the invitation to join JGC. So you don't need a specific form, just apply for the card.

I have to say JAL Card tend to go in to subatomic detail when verifying applications, so be very meticulous. And prepared for a few phone calls. I don't know if it is just for us gaijin they are that detailed or all applications. If the later they must have an army of staff, since they are pretty quick.