Decide how much of a car your production actually needs. Does your production actually need a whole car? Some productions can get by with using only half of a car shell.
Contact junkyards, high school auto body teachers and used car lots for inexpensive car portions to find what you are seeking. Explain to them that you are only wanting to use just the shell of the car. Sometimes when they realize you are wanting this for a production, they will donate it for free and will even help you get it on and off stage, on a volunteer basis, in exchange for production tickets.
Once you find your car shell, use rolling flats to move it back and forth on and off stage, to protect the stage from scratches. These can be concealed by the tires and by painting them black, where the audience cannot see them from where they are sitting, if you paint the entire flat and the wheels entirely black. This method should help with movement of your car, also.