After opening a support ticket with Veeam, we've discovered that the System Public Folders are stored in the Public Folder Database, but for some reason their software doesn't back it up. There is a size discrepancy between the PFDB in my environment and the PFDB I can restore from after a backup. I'm missing 130MB in the backed up version.
We discovered that the path of the System Public Folders root can be found in EMC by going to the Public Folder Management Console and left-click System Public Folders, then EFORMS REGISTRY. In the right pane, right-click Organization Forms and go to properties. It shows on the General tab the location of the object (Public Folder Database:).
So I know it's in the database, but Veeam doesn't process it during backups. I made a manual backup of my PFDB and used StorageCraft GRE to open the database and the System Public Folders were there.