Is a marriage considered valid if it takes place only through the offer and acceptance between the man and the woman, without witnesses or official documentation, with the knowledge and blessing of the wife's family over the phone?

This marriage is invalid by the consensus of the four schools of thought. Her saying, "I accept your marriage according to the Sunnah (definition) of Allah and His Messenger," does not make it legitimate, because it took place without a guardian (wali) or witnesses; rather, it is fornication and adultery. You must separate from her immediately. If you wish to marry her, the legal pillars of the contract must be present.