The answer to your question really depends on what your parenting plan says. Does the parenting plan outline how your daughter is to be brought to you for the visitation? If it states you are to go get her and bring her back, then your ex could rightfully stop your daughter from travelling by herself. I suggest you consult with a family law attorney in private to see what your best course of action is.
Definitely seek counsel. Have the attorney review the Parenting Plan as well as the order for child support. When there is an issue regarding payment for the child's transportaion, the courts usually order that the parties pay for it in the same ratio as the order of child support shows for the percentage of their income bears to the total income between the two of you (net income). You may want to have the court determine the issue of traveling unaccompanied. Just because some airlines allow it does not necessarily mean that it is suitable for your child.
This post is not meant to be legal advice for a specific situation. Additionally, it does not constitute an attorney client relationship. It is always recommended that you consult with the attorney of your choice before taking action which can affect your legal rights.