My daughter is getting a divorce, she lives in KY and will be moving to live with her parents because she can't afford to stay where she is. The two adults are talking about divorce but have not filed yet. Can she leave the state with her child?

If neither have filed for divorce yet, yes. However, if the father files for divorce before she can establish residency and file, then jurisdiction will be in KY. And she might be forced to give the father custody or return to KY.