Learn English – Difference between “not any” and “no any”

If I did not have a car, I would... is correct here. If I had not a car is not, because you aren't negating the right element. "a" is the indefinite article, but it also counts 'one' in a sense. As such, you would need to negate 'a':

If I had a car, I would drive to the store.

becomes

If I had no car, I would walk to the store.

However, in English you would more commonly say "did not have a car", because negating "have" makes more sense, and so "did not have" places the emphasis on that part of the sentence. You could also say "If I did have a car," which would emphasize that also, but we don't tend to in my experience. I'm not entirely sure why the distinction, but "If I had no car" sounds odd to me despite being technically sound, and "If I did have a car" also sounds a little off.

Both some and any are used with indefinite reference.

Some is used if the idea is restricted or limited in some way. Any is used if the idea is unrestricted or unlimited. Any applies to all or none; some applies to part.

The restriction may be a real one - There's some cheese in the fridge - or a psychological one, existing only in the mind of the speaker - Would you like something to eat?

Michael Lewis (1986.34), The English Verb, LTP: Hove.

Do you have any wine/water? - Unrestricted

Can I have some wine/water, please? - Restricted, a limited amount