Is a sick person who is terminally ill and prevented from fasting required to feed [someone] from what she eats, or is it permissible for another person to feed [someone] on her behalf, and is it permissible for this person to feed her family, consisting of 8 children and the husband, and is this expiation sufficient for her?

If a woman suffering from a chronic illness is unable to fast, she is permitted not to fast, and she is obligated to pay a fidyah (definition) (expiation) of feeding one poor person for each day she misses. If she is unable to feed the poor, the fidyah is waived for her. It is permissible for another person to volunteer to feed on her behalf, as it is a financial act of worship that accepts substitution.

A man came to the Prophet, peace and blessings be upon him, and said: "I am ruined!" The Prophet, peace and blessings be upon him, asked him: "Why?" He said: "I had intercourse with my wife in Ramadan (definition)." The Prophet, peace and blessings be upon him, said: "Give charity." He said: "I have nothing." So he sat down. Then a man came, leading a donkey with food. The Prophet, peace and blessings be upon him, said: "Where is the one who is ruined?" He said: "Here I am." He said: "Take this and give it as charity." He said: "To someone poorer than me? My family has no food." He said: "Then eat it."

It can be inferred from the hadith (definition) that if the person on whose behalf the food is given is poor, it is permissible for them to take what was given on their behalf. Therefore, paying the fidyah for this woman is valid, and it is permissible to give it to her as long as she and her children are poor.