Get an application through the Labor Board website for an application. All legitimate applications will be on sites with a URL ending with .gov. The form must says it is for entertainment work. A regular work permit for a child is different than an entertainment work permit.
Download and fill out the form. The form will need to be signed by a doctor or a teacher depending on the child's age. A copy of the child's birth certificate must be attached to the application.
Mail in the form and keep a copy for yourself. Include a self-address stamped envelope, so that it can be mailed back to you. It may take the state three weeks to return the form.
Make two copies of the returned form so the child can carry one to any job assignments and you have one at home. The studio teacher will need to see the work permit and sign the back before your child can work.