Is the sister considered at fault for secretly giving a poor woman a bottle of juice daily, after her mother became angry and said that this act would make the poor person resent them, that her intentions were malicious, and that she was displeased with any action her daughter took without her knowledge?

If you are buying juice with your own money for charity, this is one of the best acts of righteousness that Allah loves, and your mother has no right to prevent you. You must be gentle with her and beware of angering her. You can give charity without informing her, and her disapproval of this act does not harm you.