I do think out of this group of threads yours is the most finger on the pulse.
Shadow software simply means that the organization has applications somewhere in there that they don't 'officially' account for, ie. IT doesn't recognize them, so they provide no support.
It's a specifically named topic because it's so dangerous, for the huge alarm bell reason in the middle of your own situation description - "If anything happened to that database, we would be "toast". Ie. you currently are in an agency who has mission-critical apps on fragile software with no support. That should be a red flag - Somewhere!
So while I think every one sympathizes with you, and agrees IT should empower you with tools that are the modernized, Cloud equivalent of MS Access, I also agree there is a need to modernize the whole bureaucracy as that's the rool root issue you also describe.
The right investments in the right technologies, especially those around Cloud/PaaS (Platform as a Service) are key as they address the automation (reduce server orders from 6 months to 10 mins) as well as the common apps that users need (eg 'MS Access in the Cloud' kinda thing).