"I'm a new stage manager and am very interested in seeing how a Broadway show runs!"
I actually think that would turn me off I received a letter stating something generic like that, be more specific . . . "I am a student stage manager, and interested to see a the specifics of running a large scale commercial show" - sounds better - even better do some research on what you find particularly interesting about that specific show. I mean, you should have a basic understanding on how a Broadway show runs, I means - it's not that radically different then any other show - it just has more people, more money, more toys. If you seem a bit more educated, it may open the doors up quicker - theatre is dangerous backstage, and if you come off naive and completely uneducated, I don't want you shadowing on deck.
When I did two weeks of observing shows in New York, and 8 weekends in Las Vegas shadowing on shows - the way I presented it was, I was about to graduate and was I interested in exploring options for career goals. I used connections from school, co-workers, etc, to get me personal introductions, and then . . . as I cold-contacted other stage managers, I said, I have already set up observations on show X, Y and Z, and hoping that I could fill my time with shadowing on your show . . . it was pretty quick to fill up my dance card - I was in town for 14 days, I think I shadowed on 18 shows, and met stage managers form other shows. I never used as a job search, I never dropped of my resume, never spoke to them like, "Hey, I am graduating in six months, hire me." I just honestly keep my eyes open, ears open, asked good questions, and used it as the tremendous learning opportunity it turned out to be.
Another side hit, I actually find it better to shadow on a show you have seen - - - it keeps you focusing on the backstage stuff, rather then the interesting onstage stuff. It also allowed me to try to figure out how things happened backstage when I watched, and then see how close I was when I actually see it it.