Is it permissible for a man and a woman to appoint someone to conduct their marriage contract on their behalf while they are absent, and for the contract to be done over the phone, especially since the woman is not a virgin?

A marriage is not valid unless its pillars are met: the guardian (wali), two witnesses, the place (mahall), and the صيغة (formula/pronouncement). Authorization (tawkeel) in a marriage contract is permissible for the man, the woman's guardian, and the woman, meaning it is permissible to authorize someone to act on their behalf in making the offer (ijab) and acceptance (qabul), even if they are not present. Ibn Qudamah mentioned that the Prophet, peace and blessings be upon him, authorized someone to accept a marriage proposal, giving the example of Umm Habibah. Authorization can be done over the phone, but the marriage contract itself cannot be concluded over the phone, as ruled by the Islamic Fiqh Academy.