I agree with Tia Lydia. The situation is the same in the UK. I recommend avoiding "no car", although you may see it in certain situations. If someone asks you e.g. for a lift home from a party, then say: "Sorry, I don't have a car." If you are referring to the absence of a singular thing, then it is natural to make the auxiliary verb negative. If someone says to you. "I have 10 cars. How many do you have?" Then it would be normal to answer: "I don't have any cars" or "I have no cars" (a bit more formal). These forms of words apply when the situation refers to plurals or uncountables (e.g. I don't have any money / I have no money)

Oh wow. This is really tricky. You could say any of those choices and be understood. I think the most natural-sounding to me is "I don't have a car." "I have no car" is correct but a little formal. I wouldn't use the plural form or the "any" adjective unless you are talking about selling or loaning a car from a number of availavble vehicles. That is my humble recommendation. I am a native English speaker from near Washington, DC, USA. English speakers from other areas may have other ideas. Take care!