You do not have to be a full-fledged union member to work on a Broadway wardrobe crew- BUT you must be registered with the union (Theatrical Wardrobe Union Local #764). To register, you must have several years of experience working wardrobe, letters of recommendation from supervisors/co-workers, and of course pay a fee. I'm not sure of the exact requirements as of now, but if you check with #764 they could tell you. I don't know how many applicants are accepted, but the sure way is to already have a job offer.
Once you are working on a show and you have accumulated 30 days' work (on one production) you must officially join the union or stop working.
"What- and quit show business?" - the guy shoveling elephant shit at the circus.