Both more than likely. First the injury lawyer would get as much as they can from the company's insurance, THEN if that was successful. They could go down the path to get money from the driver, seeing how it was their fault to begin with. It's all really up to how greedy / badly injured the victim is.
Reasoning: if the car had been involved in an accident, who would be sued? Just the driver alone, or the driver and the car's owner?