What should we do, given that the father swore an oath to his wife and children that they would not speak to his wife's sister (their maternal aunt) because of a problem that occurred between them, and is it permissible to obey him in this situation?

The Prophet, peace and blessings be upon him, said: "Whoever swears an oath and then sees that something else is better than it, let him do that which is better and expiate for his oath." Thus, breaking an oath does not necessarily entail sin, whereas abandoning and severing ties of kinship is disobedience to the Creator, glory be to Him. There is no obedience to a creature in disobedience to the Creator. Mending relationships is obligatory, and the father must expiate for his oath and do that which is better.