The good news is that your mother sees you as a talented person. She may think that pushing you to perform is a way to help you overcome your shyness. Unfortunately, her good intentions are backfiring. You’ve resorted to lying, creating a bigger problem.
Not everyone is a born performer. Not everyone is comfortable enough with themselves at 14 to put themselves on stage. That doesn’t mean you have social anxiety. It may only mean that you aren’t ready. You have a lot of growing to do in the next few years.
I suggest you start working on your shyness by getting involved behind the scenes. Every show needs people doing back stage work or publicity or props or makeup. Contact whoever is in charge of the show and find out where you can volunteer. Working on a project with other people takes the pressure off being social since you are focused on a mutual project, not so much on trying to make friends.
Then have a talk with your mom. Tell her that you appreciate her concern and that you agree that you need to become more comfortable socially. Explain that you have become involved with the show in a way that will help you get comfortable for maybe performing in another year. By talking about what you are doing instead of what you can’t do, you may be able to have a more positive conversation.
I wish you well.