Is it permissible to give a family, who has a mentally disabled daughter and wants to buy a used car to facilitate her transportation, from zakat money to cover the remaining cost of the car?

The basic principle is that it is not permissible to pay zakat (definition) for purchasing a car, because it is a need and not a necessity. An exception to this is if the condition of the disabled daughter necessitates transporting her between clinics, and this cannot be done by ordinary means of transportation except with excessive difficulty.