Some of the major vintage car insurance companies will not cover a vehicle if it is in a "for hire" carrier, that is why it is important that the transporter provides legitimate proof of insurance. Your insurance company will cover your car if you are transporting it but if you give your car to a for hire carrier, the carrier is expected to cover the insurance-after all, the car is in their possession and if they are a responsible carrier, they should be responsible for it. The larger companies such as Passport and Intercity provide this and will provide written proof of this.
I agree, as soon as you pay for a car, you should get insurance for it. However, as a buyer, you need to be aware who will be responsible if something happens while it is being transported by a hired carrier. I sell A LOT of cars-many of which are very high end-and I know this to be a fact. I have been doing this for decades.
Bottom Line: Please do your homework.