Islander_fan said: "To be honest, no show has just one stage manger. There is a rotation."
There is a team, made up of a Production Stage Manager, Stage Manager, and Assistant Stage Manager(s).
Typically Stage Management receives these kinds of requests and will follow through on them only out of the goodness of one's heart. It's absolutely not their job to handle fan mail/requests; but if requesting a cast-signed Playbill, it would be best to include a check to BC/EFA and a self-addressed stamped envelope.