The preferred form is : we had people asking questions. Native speakers debate the use of had and got. To avoid confusion, I suggest you say : We caused people to ask questions. Another form is : We made people ask questions.
Have / Had is a posession verb. Had means you posessed some thing or some quality. In your example, the behavior " shooting on location " is done by the subject "we". Thus, "we" posesses the quality that makes people ask questions.
Got / Get is a verb that means to take posession. Get the book. I got the paper yesterday. The verbs had and got are similar. Many people confuse them.
I prefer make and cause verbs. They are similar. Each one means the subject creates an effect on the object. I made you happy. I caused you to be happy. These mean the same thing.