Are there any companies that you have gone back to a number of times? If so, you could include your supervisors/etc from those positions, and then have the more complete list on your resume. If you have those as several examples from your career experience, and more details in your cover letter/resume/interview to flesh things out, that should give them a good idea of what your experience is without listing 50 shows.
I had to do the same thing when applying for my current position- fitting a theatrical-style resume into a traditional work history web form is a pain! I chose a couple of examples, and then included freelance PM work and events work as different positions, since they use different skills. I really tried to tailor my experience to the job posting, so if you have any experience that really lines up with what they are looking for, those might also be good shows to use for this web form.
Good luck!