Is it permissible to marry a young man whom the family refused to approve due to the brother's disapproval, bearing in mind that the relationship with him was limited to phone calls, and that the girl feels at ease only with him and strives to please Allah through her efforts in what is lawful?

It is not permissible to marry the aforementioned young man without a guardian (wali), for the guardian is a condition for the validity of the marriage contract (nikah (definition)), and without him, the marriage is invalid. The guardian in this case is the father, not the brother. It is advisable to seek the intervention of good-hearted people to persuade the father to agree to the marriage. If he agrees, then the marriage should proceed, and the brother's objection is irrelevant. It must be noted that interaction between men and women, whether online or otherwise, is forbidden even if it is for the purpose of marriage, because whoever wishes to marry must propose to the woman through her guardians.