How old do you have to be to work at child time day care?

I do not know. You should ask the state authority that licenses day care facilities. They make the rules.

I had to have written permission from the High School to permit a 15-yr-old to work a part-time job in the law office, so while a 16-yr-old can work, there may be many oppressive state and county regulations that force employers to just say 'skip it'.