If you have an auto accident on company time, whether your employer is "responsible" depends on what arrangements have been made in advance and what you mean by "responsible."
Were you driving a company vehicle?
Were you engaged in company business or on an errand for your own benefit?
Were you complying with company policy regarding distracted driving, hands-free cell phone use, etc?
If you were driving your own vehicle on company business, was that a routine process and had you notified your insurance company of that practice?