I'll answer the question in the title:
"Amateur" theatre groups are commonly called community theatre groups. Anyone who wants to put on a musical, whether it's on-Broadway, off-Broadway, touring, out-of-town, community, school, etc., they all have to pay to license the musical. One of the protections granted by copyright law is the exclusive rights to perform the work, so for anyone who is not the copyright holder, they have to pay for a license.
Licensing is pretty tightly controlled for musical theatre, which I think is because of both the need for the authors of the musical to recoup as much as they can of the investment of time they spent writing it, as well as to make sure there aren't low budget productions competing with more lucrative productions in the same area. For example, right now Hamilton is earning so much money with the Broadway and high-budget touring shows that the licensing fees are certainly very large - to prevent community theatre groups from performing it all over the place for a fraction of the professional ticket prices and saturating the market. If you want to see Hamilton right now, you pay through the nose, and that's largely due to the licensing strategy. After interest in the show starts to wane and the Broadway run closes, licensing costs will come down to make it more accessible to smaller and smaller groups to be able to keep at least a bit of money coming in for the copyright owners.
Buying a license to perform a musical includes both the rights to perform it and the materials necessary to do so, which isn't always limited to just the score and book and parts. For instance, a license for Avenue Q can optionally include copies of the puppets used on Broadway.
After the run, all materials must be returned in their original condition. There's often an "erasing party" for the orchestra at the end of a run where the players go through and erase all the pencil notes they have made in their scores. It's illegal to copy the materials although some musicians memorize their parts and might jot down notes to remember how to play it.
If you want to start playing musical theatre works, the best thing is to get in with a community theatre group. As I mentioned in my comment, you'll probably have to offer to have your own braille transcriptions made, or take care of any other accommodations you might need. I'm sure another orchestra member or cast or crew member will be happy to help with other needs, such as getting to your place for rehearsals and shows, etc. I have generally found musical theatre people to be lovely and kind (a few stressed out producers and directors notwithstanding). In many areas, community theatre groups have a hard time finding capable musicians, especially for certain instruments, so you might not even have to audition.
Your only other option is to get a piano/vocal book from a music shop and try to work out the part for your instrument or what is different from the actual score if you are a pianist.