I figured out what I did wrong. I had a free app assigned to the iPads, but I did not have the app added to the app catalog so that I could use VPP to deploy it. Once I added it to the App Catalog and set it to use VPP, I no longer got the prompt.

That solution did not work for me - unless I'm not understanding where the app catalog is in Casper. I have the device-assignable free app added under Mobile Devices>Apps, properly scoped, but the request for iTunes still appears on the iPads.

That is the correct location, but on the VPP tab for the app do you have Assign VPP Content checked?

I had the same symptom a few months ago. It was due to a free app not being added / deployed via VPP.

This prompt pops up for me when the following is enabled...even for supervised devices

Make app managed if currently installed as unmanaged
Manage the app if it is currently installed as an unmanaged app (the user is prompted to allow management on unsupervised devices)

Go to Mobile Device Apps > select app > Managed Distribution > Assign Content Purchased in Volume.

Fixed!

This does not fix it for me. I'm constantly facing this problem and still haven't found the source of the problem. 

Same. All of my six apps are free and are distributed through Apple School Manager via VPP. I have "Assign Content Purchased in Volume" for all of them. Still getting this annoying pop-up.