Let's say I had an exact number of tasks at the beginning of a day, and I forgot in the evening how many I had when the work day had started. What is the correct or more casual way to phrase that question? Is "How many we had this morning" correct?
FF provided a good answer in the comments, but I'll expand a bit.
You have a nominal clause (also called content clause, noun clause, etc.). These clauses begin with an introductory word or phrase (in this case, "how many", which functions as the clause's direct object), and then the rest of the clause follows in typical word order: in this case, first the subject ("we"), then the simple predicate ("had"), and then the adverbial phrase ("this morning").
A nominal clause is a type of subordinate clause, so it can't constitute a complete sentence (in the traditional sense) by itself. To make this into a main clause, we can invert the simple predicate and subject, thus making the sentence interrogative:
How many had we this morning?
However, this kind of question is archaic. In modern English, the simple predicate must be the auxiliary verb "to do". The verb "to have" then becomes its complement and appears in the bare infinitive form:
How many did we have this morning?