Can I sue a car dealership?

The answer is "no, you cannot sue the dealership".  Car dealers are not required to make sure their customers have insurance.  Your only remedy may be to sue the woman who hit you.  Just because she didn't have insurance doesn't mean she can't be made to pay you for the damage.  First, you need to check your own insurance to see whether your policy has what is called "uninsured motorist coverage", which is exactly what ot sounds like - it pays for your damage if you're hit by an uninsured motorist.  If you don't have uninsured motorist coverage you need to consult a local lawyer who can investigate and find out whether the matter is worth suing over. WARNING:  I am not your lawyer and providing you with the generic information, definition, clarification or explanation you seek about a legal matter on this anonymous forum does not make me your lawyer.  By answering your question I am only trying to provide you with general information that will deepen your understanding of the legal concept that appears to be an issue for you, that will hopefully guide you in selecting and communicating with a lawyer of your choosing.  The reason for all this is that legal problems are intensely fact-driven, and it is impossible to give anyone correct legal advice without comprehensive inquiry by the lawyer into the specific facts of the situation that concerns you, and the law of the jurisdiction in which you reside, and that is impossible to accomplish through this website.  Even though I am not your lawyer, if I should later become your lawyer no representation is made that the quality of the legal services to be performed is greater than the quality of legal services performed by other lawyers.