Is the father's and sister's promise to the female student to take her to Umrah upon her success and admission to university considered a vow (nadhr), and is it permissible for the paternal aunt to expiate on behalf of the father or vice versa if it was a vow and the father fulfilled it?

What your father and aunt stated is not considered a vow (nadhr) because it does not include the specific formulation of a vow. Rather, it is a promise, the fulfillment of which is not obligatory. However, it is recommendable according to the most preponderant opinion among scholars, and it is better to fulfill it.