Profile Manager not sending email invitations
When you acquire VPP app licenses for free applications, are they appearing in your Apps tab in the Profile Manager [device manager interface] sidebar?
Have your devices successfully enrolled to Profile Manager, and can you deploy configuration payloads/profiles to them?

We ran into this when getting OS X Server and PM up and running, and then realized you can distribute apps and books without requiring the user to approve these VPP invitations.
Unless you're attempting to deploy internal/enterprise apps, you can deploy App Store/VPP apps to devices or device groups as long as they are enrolled to Profile Manager (supervised or not, they should deliver as long as your config profiles are).
EDIT: You should also make sure that you have an APNs account and that APNs is enabled in your OS X Server, with an updated token.

If you‘re not getting Apple push notifications - https://support.apple.com/en-us/HT203609
Resolve issues with Profile Manager in macOS Server -  https://support.apple.com/en-us/HT201063