Praise be to Allah.
It is permissible for a person to delegate someone to distribute his zakaah to the needy, on condition that the person appointed is honest and trustworthy and will give the zakaah to those who are entitled to it. There is no differentiation in this regard between men and women, so it is permissible to give zakaah money to a trustworthy woman who can give it to those who are entitled to it.
The basic principle concerning that is that it is stipulated that the appointed person be one of those who is allowed (in sharee’ah) to handle for himself the matter that is entrusted to him. A woman may distribute her own zakaah, so it is permissible for her to be appointed to do that on behalf of someone else.
Ibn Qudaamah (may Allaah have mercy on him) said: If a person is permitted (in sharee’ah) to handle anything on his own behalf and it is something in which another person may be appointed as a proxy, it is valid for that person to be appointed to do that thing on behalf of others, whether that person is a man or a woman, free or slave, Muslim or kaafir. End quote from al-Mughni (5/51).
See also Nihaayat al-Muhtaaj (5/18).
And Allaah knows best.