Any private company (or their agent) is free to determine the conditions which apply to parking on their land. IF THEY DISPLAY CLEAR SIGNS stating the charges for parking on their car park (or on a particular part of the car park, such as in disabled bays) anyone who then parks on that car park automatically enters into a legally binding contract to pay the relevant charge.
So, IF THERE WERE CLEAR SIGNS, your daughter entered into a contract to pay the relevant charge for using a disabled bay without a blue badge.
There are plenty of people (posting here on AB and elsewhere) who will try to tell you that the parking charge is unenforceable. If you (and your daughter) listen to them, your daughter could find that she faces a much bigger bill:
http://www.rochdaleon...alty-in-landmark-case
Chris