What do you call someone who performs in musical theatre?

Artiste = a skilled performer, especially a dancer, singer, or actor

Cambridge dictionary

It is often more difficult to go from a specification of meaning to a relevant word than to go from word to meaning, which may be done by dictionary search. Nevertheless, I suggest that you could have done this by looking at synonym lists for dancer, performer, singer, actor. artiste would have emerged as a possibility.

I give this answer and advice because you are a welcome newcomer to the site. Others may close the question because you gave no evidence of your own research before posing your question.

The catch-all word would be performer. To be more specific would depend on the show and a performer's on-stage role. In an opera the performers are definitely singers. In most musical plays they are actors, even though they must usually sing and dance. Some musical plays have roles specifically for dancers.

As a professional description most performers in musical plays would probably describe themselves as actors. A performer could also say "I was in Oliver in Melbourne in 2003", without specifying more precisely.

Apart from calling them just a singer or dancer, if they did all that and more you would call them a variety performer. You would be very surprised to see how many people known for just one skill are so flexible.