Can I sue a big company whose driver hit our car in a drunk driving crash?

The issue is whether the driver was working at the time of the crash--e.g. traveling to a client or meeting, making a delivery, etc. If she was on the job at the time, her employer may be liable for her actions (particularly if they had any notice at all of any drinking problem, or she'd left a company event where alcohal was served, etc.). On the other hand, if she was not working at the time, the company would NOT be liable for her actions--e.g. if she stopped off a bar or party on the way home after work and happened to hit you, the company is not liable and would not have to pay you anything (though they may choose to, to avoid bad press).

There is no easy answer to your question--it depends, as you can see, on the circumstances, and also on the extent of your injuries, damages, or losses. The best thing would be to meet with a personal injury attorney (many or most provide a free initial consultation to evalutate your case) and discuss your situation in detail.