I believe this will be possible.
The GitHub API caters for the management of Forks.
What you could then build is a web based document management app, that runs on one website(server) with a MySQL database for handling instances of a document and users, sharing rules, permissions etc. You could then use GitHub as a storage engine for all documents(repositories) and it could take care of versioning, roll backs etc.
You could then build in a Markdown editor, so all of your documentation could be written in markdown, and then stored on GitHub.
Doing it this way, you would also not need to run git or do, push or pull requests on your local machine as all of those requests would be running server side within the app.
This would make, working with non-technical staff, a bit easier plus it would mean you could run this on lower end devices without having an impact on the performance of the device and user experience.
You could eventually build an API for the app and extend it to support iOS and Android versions with offline support, so you could actually do all of your documentation in ‘offline mode’ and then sync changes to your app later, which would take care of the versioning etc.
Here are some resources to take a look at. 🙂