Is it permissible for one of a girl's brothers to solemnize her marriage contract—after her eldest brother refused to do so to appease his mother—knowing that all her brothers are mature and qualified to solemnize the contract, and can the contract be completed by phone, or is a proxy required?

If a woman has more than one guardian (wali) who are equal in kinship, such as brothers, any of them may عقد (solemnize) her marriage. It is preferable to give precedence to the eldest and most virtuous among them. If one of them marries her to a suitable man (kufu') with her consent, the marriage is valid, and the remaining guardians have no right to object. You may also appoint someone as your agent (wakeel) to solemnize your marriage, or you may solemnize it yourself via a phone call in the presence of witnesses.