Hi Ben,
> I have made a web map and web mapping application that I would like to take offline and share strictly within the local network of my workplace. Is this possible and if so how would I go about doing this?
No, this workflow isn't really supported. The point of a web map, is that the data its referencing, typically web services, is accessed "live" in in real-time. And a web app is designed and meant to run in a web browser.
If you own the data sources powering the web map, you could download those locally - then perhaps access them in ArcGIS Pro or ArcMap. The help topic to download feature layers locally,
Extract Data—ArcGIS Online Help | Documentation
Hope this helps,
