Not unless the car was in the business name, or an employee was driving it at the time in the course of business. There has to be SOME connection between the accident and the business, you don't get to sue a business just because the owner was involved in an accident. There are a myriad of ways to do it, however, depending on all kinds of factual scenarios. My guess is she is bluffing.
This is general advice and does not establish an attorney-client relationship.
I agree with my colleague-- unless the business has a connection to the collision, there is nothing to worry about.
In no way am I offering you legal advice, and in no way has my comment created an attorney-client relationship. You are not to rely upon my note above in any way, but insted need to sit down with counsel and share all relevant facts before receiving fully-informed legal advice. If you want to be completely sure of your rights, you must sit down with an experienced criminal defense attorney to be fully aware of your rights.
It is unclear to me why they are threatening to sue you at all. It appears you gave them more than what they paid for the car, and they were at fault for not changing title to it quick enough. Why are they unhappy? Do they want the totaled car price in full from you?
licensed attorney in Montana. Your specific state laws may be different.