Hi Christine,
Thank you for visiting the Disney Parks Moms Panel with your question. My daughter has a hearing impairment, so I think about the best way to plan for and navigate the park with a hearing impairment often.
You can book Chef Mickey's online. Just visit the
Disney Dining Reservation site for Chef Mickey's. Using the "check availability" square on the right-hand side of the screen, you should be able to check availability for the dates in which you are interested and then book online. You can make any of your
dining reservations for Walt Disney World through this site. Using the Disney online dining reservation system, I have never had the experience of it telling me to call a phone number. If you are, I would suggest both trying again at a later time to see if it resolves itself.
Another option if you have access to a TDD is to use it to call 407-939-7670, which is a TDD line that will connect you with Walt Disney World using the TDD.
I hope that this helps to answer your question, Christine. Please do not hesitate to ask if you have any further questions.