Is it permissible to give my sister-in-law (my brother's wife) from my Zakat money to buy school supplies for their children, with or without my brother's knowledge, given that he is of moderate means, has four children, and cannot meet their school needs?

It is not permissible to give zakat to one's brother or his children if they are not poor or needy, as their maintenance and sustenance are obligatory upon their father. Zakat is disbursed to the eight categories mentioned in the Quran.

If the brother is unable to cover his children's educational expenses, it is permissible to pay zakat to purchase books and educational necessities for his children. This has been allowed by some scholars, such as Ibn Taymiyyah. Zakat can be given to their father or mother for this purpose, and it is not required to inform them that it is from zakat.