Learn English – Why do we not use these helping verbs (do, did and does) in positive sentences

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.

SHORT ANSWER:
The simple present and simple past are by definition verb constructions which are not accompanied by auxiliaries (helping verbs). That's why they're called "simple".

LONGER ANSWER:
First, a terminological note: When I speak of "tense" below I don't mean the many single- and multi-word verb chains which English teachers and students often call tenses. I call these "constructions". By "tense" I mean the past or non-past time reference expressed by inflection: a change in the form of the verb, such as adding an ending or changing the pronunciation.

Do, does and did are used in present- and past-tense affirmative sentences; in fact, the principal use of this construction is to emphasize that a response is affirmative:

A: You never listen to me.
B: I do listen to you! I'm listening to you right now!

A: Brian doesn't like my cookies.
B: I think he does like your cookies, but he's trying to lose weight.

A: You should get a haircut.
B: I did get a haircut, just yesterday!

However, do/does/did are not used with finite forms of the main verb: forms which express and are inflected for tense, person and number. No auxiliary is ever used with a finite form, only with non-finite forms which have no tense, person or number.

• The auxiliaries have and be are used with participles of the main verb:

  She has broken my heart. (broken is the past participle; HAVE + past participle is the perfect construction.)

  My heart was broken when she left. (BE + past participle is the passive construction.)

  She is breaking my heart. (breaking is the present participle; BE + past participle is the progressive construction.)

  Note that with most English verbs—'regular' verbs—the past participle (broken, here) and the past-tense finite form are identical, which leads many learners to think that the auxiliaries are used with the past-tense form. But so many English verbs—'irregular' verbs—have different past-tense and past participle forms, and those verbs are used so frequently, that we have to make the distinction them in order to make sense of how the language works. —Hold on to that thought; we're going to come back to it shortly.

• All other auxiliaries (do/does/did and the 'modal' auxiliaries can, could, may, might, must, shall, should, will, would) are used with the infinitive form of the main verb.

  This, again, is difficult for learners to recognize, because with every verb except one the infinitive form is identical with the plain present form. ...

Then why do we give a different name to the form we use with do/does/did and the modal auxiliaries? —for two reasons:

1. Unlike the many English verbs which have distinct past participle and past-tense forms, there is only one verb which does have a distinct infinitive form. But that verb is be, which is so central to the language that it can't be treated as a minor exception. Note that we never use any of the present- or past-tense forms with the modal auxiliaries or with the 'infinitive marker' to:

   Jack will be here tomorrow, not *Jack will is
   Jack had to **be* in London yesterday, not *Jack had to was

   Consequently, we distinguish the infinitive from the identical plain present-tense form for the same reason we distinguish the regular past participle from the identical regular past-tense form: because it lets us make sense of how these constructions are built.

2. Historically, the infinitive and the present form were different, marked with different endings. Those endings were lost in the course of the transition from Old English to Modern English, but there are still 'traces' of the difference in English syntax. Once more the distinction of identical forms enables us to recognize important rules of the language.

   For instance, there is the rule that in any verb construction only the first verb in the chain can be a finite form; all the rest must be non-finite forms.

   That's what's happening with I do listen and He does like your cookies and I did get a haircut: the first verb, a finite form of DO, is inflected for tense, person and number, and it is followed by the infinitive form of the main verb, listen, like, get.