Has paternity been established with the father?
If neither parent officially has custody on paper then mom could probably do that, but it is extrmely ill=advised if she is going to take the baby out of state. Dad's state currently has jurisdiction and dad could easily file for custody and ask for the child to be returned. Mom would not be able to file anything until she has been in this new state for at least 6 months and that would give dad ample time to file paperwork.
Not to mention, that might irritate a judge.
Has mom or dad filed any petition for custody or visitation?
It sounds like Dad has been the primary caregiver so it is not automatic that your daughter is going to get custody. She can certainly file and either start a full blown custody battle, or try to work something out with dad.
If mom wants custody and wants to move out of state, she should file within the courts and ask. Be advised though that if mom is allowed to move out of state with the child, she will probably have to provide the bulk of the transportation for visits.