What is the religious ruling and what should I do if I vowed to send a relative for Umrah, but he refused to go after what I vowed for had been fulfilled?

If someone makes a vow of obedience, it becomes obligatory for them to fulfill it. If the vow is made to a specific person, the majority of scholars hold that this designation is binding, and it is not permissible to direct it to anyone else. If the one who made the vow wishes to fulfill it, but the person for whom the vow was made refuses, then there is no blame on the one who made the vow, because they did what they could, and the refusal came from another.