Articles belong to a group of words called "determiners". Besides articles, there are other determiners in the English language, and "no" is a determiner too.
Let me quote from BBC:
No is a determiner expressing quantity like 'all', 'every', 'many', 'some', 'any', 'each', 'either', 'one', 'another' and is used before singular and plural nouns. It is similar in meaning to 'not a' or 'not any' and is often our preferred choice if we want to give emphasis to what we are saying.
A singular countable noun like car should have some determiner before it, and no suits the purpose. No additional determiner is required.
Indeed, in some cases we do combine not with a, I'm not sure what the grammar says about this. "I have not a car" sound quaint and wrong in modern English. Maybe several hundred years ago it was more acceptable?
In certain constructions not a + singular count noun certainly remains in use:
Not a shirt on my back,
Not a penny to my name,
Lord, I can't go back home this way.
("500 miles", a song)
I don't recall the rule explaining this.