We saw the Apple School Manager (ASM) has the Download Token section available in its Settings > Apps and Books area which led me to believe this is also for the JAMF/JSS VPP section; however, we currently have hundreds of thousands of licenses in the old iTunes Store VPP and have this iTunes Store VPP server token connected to JSS.
Are we able to simply add the new Apple School Server Tokens from the Apps and Books area and have those work fine alongside the old iTunes Store VPP or will there be issues?
Don't do any token downloading from ASM yet! Check out the "If assigned licenses don't transfer" section of HT208257.
You will want to do a migration of your iTunes VPP account to Apps and Books. The current VPP token will be associated with a Location, converted to an Apps and Books token, and then all your licenses will be usable in that location. You should not need to do anything in Jamf Pro if I remember correctly.
You might want to open a ticket with Jamf Support, and also check out this comment.