Yes, there are many situations where the form of the verb is 'have' - the base form - following 'he', 'she' or 'it'.
In fact, the only time the these pronouns are followed by 'has' is in a present simple affirmative statement: He has/ She has/ It has
In all other cases, these pronouns are followed by the base form, or unconjugated form, which is 'have'.
If you are using an auxiliary verb (does) or a modal verb (can/may/will/would/must etc) in either a negative statement or in a question, you use the base form of the verb. Here are some examples:
Does he have a car?
Did he have a car?
Will he have a car?
Could he have a car?
He doesn't have a car.
He didn't have a car.
He mustn't have a car.
He won't have a car.
...and so on.
I hope that answers your question.