Is my mother sinful and is there a make-up upon her after she broke her fast due to feeling dizzy and extremely tired while fasting in Ramadan?

One who breaks their fast in Ramadan due to a legitimate excuse—such as illness or travel—bears no sin, but must make up the days they missed before the next Ramadan if their excuse is no longer present, as Allah Almighty says: "But whoever among you is ill or on a journey [should fast] a number of other days."