Is it permissible for a Muslim who vowed to slaughter two sheep to send their cost to his brother in his home country for him to slaughter them, and is it permissible for the vower's family to eat from them, or is it better to pay their cost to an Islamic relief organization or to give them as charity to his needy relatives?

You must fulfill the vow by fasting five days and slaughtering two sheep. Paying money instead of slaughtering is not permissible. You can slaughter the sheep yourself or appoint someone to buy and slaughter them on your behalf, whether in your country of residence or your home country. It is permissible to give the meat of the sacrifice to your poor brothers, and there is no objection if they give some of it to their mother if she is poor. However, it is not permissible for you to give some of it to your mother because her sustenance is your obligation.