First, the Fourth Amendment protects against government searches, not private companies looking at data in vehicles that they have paid for. Second it only protects against unreasonable searches. We might have a disagreement about what is reasonable but I think that an insurance company to which I pay premiums might reasonably want to verify that any claims that they pay out were for "accidents", not illegal or stupid acts of the person who caused the problem.
I would suggest that if the culprit who was doing something wrong when he or she crashed, always has the ability to refuse to submit a claim and does not need to transfer title of the wreck to his or her insurance company. Then the insurance company does not buy the vehicle to gain access and I don't have to pay with my increased premiums..