Is it permissible to give the amount of a vow designated for the children of a deceased brother to his needy brothers to establish a business, taking into account that the deceased brother's children are already provided for, while the needy brothers are in greater need?

If you vowed to allocate money specifically for your brother's children, you are not permitted to spend it on anyone else, because it has become their entitlement. The Shafi'i jurists have stipulated that it is not permissible to spend vowed money on anyone other than the designated poor person, and that person has the right to demand it because it has become their right. What is obligatory upon you is to disburse this money to your brother's children for whom you made the vow. As for your other siblings, you can assist them with other forms of charity.