You can sometimes find the email address by searching various publications, but I've honestly never been too successful that way.
The most success I had was a nicely typed professional letter which I put in an envelope and addressed to the stage manager. I walked to the stage entrance and gave it to security and asked them to deliver it.
In my letter I introduce myself, told them about my goals, and gave my cell phone number and email address. I asked if shadowing was possible and asked them if they would contact me and let me know. Most contacted me via email or text and we set it up relatively easily.
I also said I'd love to have coffee with them if shadowing wasn't possible. (Some companies may not allow shadowing)
It usually took one to three days to get a reply. I'd suggest dropping off the letter early in the run (But not the first or second day, those are crazy busy for a SM and you may get lost in the shuffle)
Be professional and polite and you will be more successful than you might expect.
Chad
I too am interested in shadowing a Broadway SM while I'm doing my internship in the city this summer and I've been having difficulty finding the email addresses for the theaters/PSMs. Does anyone have any advice on how to gain this information?