If you call the theatres they will say no. They really can't be bothered.
But try the NYC Tourism Dept or some similarly titled group. Maybe even the Times Square Business Improvement District. I seem to recall at some point there was a private company that did a few backstage tours.
The New Amsterdam is a good idea and also there is always the Music Hall. Lincoln Center gives tours, but I don't know if they go backstage or not.
"If my life weren't funny, it would just be true. And that would be unacceptable."
--Carrie Fisher