Can a staffing agency require you to carry your own liability insurance if they place you in a driving position for a company that already fully cover their vehicles?

Sure they can. If you don't like the requirements of the job then don't take that job. They are probably going to insist that you carry a limit if liability equal to their limit as well as the requirement to carry coverage. If you have an accident you will probably be sued and they will be sued as well. In order to protect their insurance coverage they will require you to have this coverage for an accident that is your fault.