Yes, we have such a structure, and yes, it is grammatically sound. It is not causative.
Non-finite verb clauses with oblique subjects
- I have you returning the car on August 14th here at the airport.
- He has me returning the car on August 14th here at the airport.
This is the same grammatical structure as is asked about in this question and in this question.
The direct object is the clause you returning the car”. What you have going on here is a non-finite verb clause as the direct object of the main (finite) verb, and a subject pronoun in the object (or oblique) case providing the logical subject of that non-finite verb clause.
Yours is a simple S-V-DO sentence. The subject is I, the verb is have, and the direct object of that verb is the non-finite verb clause you returning the car that uses the ‑ing inflection of the verb.
It’s the very same same underlying syntactic structure that we see when we use the to-infinitive as the complement via a for-complementizer with an oblique subject:
Those are the same structures as your sentence because the infinitive clause and the gerund clause are interchangeable for this. In both cases, the non-finite verb clause is occupying the syntactic slot normally filled by a substantive like a pronoun or a noun phrase, because both those two kinds of non-finite verb clauses can be used substantively.
(They can also both be modifiers, not substantives, but that’s not what’s happening here.)
Note that this use of have is not a causative one. The causative have is when you “have someone do something” meaning that you make them do it. The causative have doesn’t take an ‑ing verb with oblique subject nor a to-infinitive via a for-complementizer, but rather a bare infinitive with oblique subject. For example:
- He had me return the car on August 14th here at the airport.
That is the sense of have that’s equivalent to make, the one that takes a bare infinitive clause with subject: me return the car.