Is it permissible in Islamic law for a sister to authorize her family over the phone to conclude her marriage contract in Egypt?

If you are certain that the caller is the woman's legal guardian (wali), then there is no objection to him authorizing you via phone to accept her marriage. The Islamic Fiqh Academy has permitted authorization via phone. It is important to be aware of the risks of online marriage, that it is impermissible for a woman to travel without a mahram unless there is a necessity, and that a relationship between a man and a non-mahram woman is not permissible.