Depending upon each company and their policy, you might have to pay a small percentage of your total shipping cost as a deposit to book the job. Ideally these day several reputable auto shipping companies do not request for deposit payment before booking a car for transport, they might however take partial payment before the car is picked up and the balance is due when its delivered.
Note: In either case, only a small deposit is required. Be wary of companies that are asking for 50% or more upfront to move your car.