My daughter's sax teacher (who's also a friend of ours) plays professionally for pantos, shows, tours etc. and yes, it's definitely word of mouth and "friends of friends". The main trick seems to be to get that first booking.
I don't think you'd get away with just one instrument either, as most orchestra pit musicians have to be multi-instrumentalists. As a sax specialist he takes a soprano, alto & tenor sax plus clarinet and flute if they're needed. That lot don't come cheap!
Good luck though - he really enjoys doing it!
As far as amateur productions go, look out in your local paper for local groups advertising auditions or just phone up local amateur dramatics groups and offer your services. I'm involved with an amateur group, who all perform for free, but we pay the musicians. I'm not sure how common that is.
P.S. I bet the West End production of Chicago is fun to play for - up on stage all the way through!