If we distribute an app through an VPP account (Volume Purchase), will the user also get a badge on the AppStore Dashboard icon, if an update is available (like with the 'normal' user accounts?)
There is a comparison table of the VPP features for the three possible distribution paths "MDM User Assignment", "MDM Device Assignment" and "Redemption codes".
The way I understand the table is that automatic updates through the App Store are available if you distribute via redemption codes OR assign the app via MDM directly to the user / Apple ID.
Only if you assign it via MDM directly to a device (no Apple ID required), it will not have a connection to the store and therefore not provide any automatic updates. However, if you have the MDM in place, you push updates through that anyway.