Is it permissible to give one's sister from zakat money for her medical treatment, given that she is suffering from a chest and kidney illness while living with her husband, and her employed children's earnings are not sufficient for her needs?

It is permissible for a sick person who cannot find enough for their treatment to take from zakat. Therefore, your son may give his aunt from his zakat money for her treatment if she does not have sufficient means. This would be both charity and maintaining kinship ties, as the Prophet, peace and blessings be upon him, said: "Charity given to the poor is a single charity, while charity given to a relative is two: charity and maintaining kinship ties."