How do you get permission to sell character cakes?
Contact the company that holds the copyright.
If you're having trouble finding out who holds the copyright, you can post the name of the character in question here and someone should be able to find it for you.
The chances for approval might be better if the customer asks rather than the baker. Also, if the copyright holder says no the customer gets to hear the answer first-hand, so any ill will would be directed at the copyright holder instead of the baker.
Thank you for the information. I have not started a business yet, but I want to get all of the information I can before I start.