What is the method for distributing the meat of a vow of obedience, and is it permissible to break the fast of those who are fasting with it in the mosque?

A vow of obedience (nadhr al-ta‘ah) is disbursed to the party the one making the vow intended. If he did not intend a specific party, then it is disbursed to the poor and needy. So, if you intended the poor and needy, it is obligatory to distribute the meat to them. If you intended neighbors, it is obligatory to distribute it to the neighbors. If you did not specify a particular party and made a general vow, then its disbursement is like the disbursement of charity: to the poor and needy. If the fasting persons you are referring to are among the poor or needy, then you can use it to provide them with iftar. If they are not, then no.