I'd like to use the posters of musicals as well as other images from
the musicals playing in NY and London on a commercial website to help
promote the shows. How do I go about doing this?  What kind of costs
could I expect? Thanks,

I'm not an official Google Researcher, but speaking from my own
experience in television, we've been given permission to use posters
and official production photographs at no charge as long as the
productions were still running at the time and we weren't saying
anything negative about them.  Whether this applies for a commercial
website, I don't know.

We obtained these images easily by contacting the publicists for the
plays or musicals, who then sent us high-res .jpgs.

Now, If you want to use a poster/photo/or other image from a show that
has closed, then it gets more complicated, particularly if the poster
has a painting or illustration or other key art.  You not only need
permission from the producers, you sometimes need to get permission
from the artist as well.  And this could mean license fees.    Or not.
  You never know.