State laws vary on if a driver's license or insurance is needed when you first buy and register a car. In some states the person or dealership selling the vehicle must verify the person is licensed and has insurance in order to give them possession of the vehicle and allow them to drive off in it while in other states this is not required.
In California a driver's license is not required to be shown to buy or register a car to the best of our knowledge. However if you do not have a driver's license it is very difficult to get insurance which can be an issue in most all states. In CA car insurance is needed for a car that is registered in the state.
In October 2006 the CA Department of Motor Vehicles started the process of suspending the registration of a vehicle if liability insurance is cancelled and a replacement policy is not submitted within 45 days, or your insurance company has not electronically provided evidence of insurance within 30 days of a registration card being issued on a newly acquired vehicle, or if you provide false evidence of insurance.
Before 2006 insurance was required when a person renewed registration so that is the point at which a person that is without insurance would have had their registration suspended in California. We have read that in CA and other states some illegal immigrants have others register their cars for them, which is misrepresentation and is against the law in many states, or may have falsified their registration.
In California it is illegal to drive without a license and insurance so even if an illegal immigrant was able to buy and register a car if they are without a valid driver's license and auto insurance for the vehicle then they are driving illegally.
For information on California or other state laws regarding the buying or selling of a vehicle and registration requirements contact the DMV of that state. If you have questions about what can be seen as loopholes in state laws you can contact your state legislative body.