Is it permissible for a father to prevent his daughter from driving a car without a legitimate Shar'i reason, claiming that it is out of fear for her, knowing that the daughter needs to drive to attend to her personal needs and those of her mother, in the absence of a safe and comfortable means of transportation?

A woman driving a car is permissible by Islamic law. A father preventing his daughter from driving might stem from his fear for her. You should try to convince him of the necessity for it and seek help from some relatives or righteous people to mediate. If he remains unconvinced, then you must be patient and dutiful to him, and strive to avoid being alone with a foreign driver.