how it typically works,is a player fills out the paperwork to join the union and in most cases has to participate in official activities (like playing in at least 1 game for sports) to be eligible to represented by the union. the hold up with a lot of these international players is there are international laws that have certain requirements before they can join unions in other countries,with ohtani I think they applied for exceptions and the parties involved negotiated so he could be included immediately (with his popularity they did it for all the merchandise sales,etc)
Yamamoto certainly seems popular enough that they could go the same route, so I wouldn't be surprised if he's there or if he is not there.