We'll see. Proof of vaccine would be revealing personal medical information, unlike masks.
And I realize schools require vaccinations, but I think that's by legislation. Will states be enacting laws about vaccines for people wanting to go to a theatre? Can this be done by mandate?
(And for the record, I'll be first in line when my number comes up to get a vaccine, and I would have no problem showing my sticker, membership card, or whatever to prove I got it - though Broadway isn't the first place I'm going.)