Is it permissible to give the wife's family from the Zakat al-Mal (obligatory charity) to equip the kitchen, after they refused to do so due to their financial circumstances, despite a prior agreement?

The wife's family is obligated to fulfill their commitment to furnish the kitchen, based on the Prophet's (peace be upon him) saying: "Muslims are bound by their conditions." If they are unable, they are excused until they can. It is not permissible for you to pay your zakat money to them to furnish the kitchen, because that would be a protection for your own wealth, and Allah Almighty has specified the recipients of zakat among the eight categories mentioned in Surat At-Tawbah.