The answer to your question is more for the Board of Education than an attorney, I think. The school only approves one week of vacation, but that does not mean they can't take her out for unexcused absences. I don't see how an attorney can make it mandatory that they take her on vacation. An attorney has no authority to do this.
You call him your ex, is she an ex-husband? Or, ex-boyfriend? The fact that there is no custody order makes me think he is an ex-boyfriend. Because he is the biological father, he does have rights to see his child. Is there any kind of order in your case? Any child support order?
This is a tough case with no order. There is no way to force anyone to do anything because there are no legal actions to be taken if no order is in place. The best you could do is have the Board of Education, or her school, tell your ex that the absences would be unexcused because your vacation has already been approved.