Is there a possible way of having a report show how much space is free on a network mapped drive, or setting up an agent procedure to query for the information?
It's generally not a good idea to do this -
1. You should be monitoring the server or equipment providing the mapped drive directly, either via agent or KNM.
2. Mapped drives are a per-user entity, not a system entity, so you are dependent on the user being logged in for the mapped drive to even to exist.
3. If you do choose to use a script, then you will need some sort of app or tool to enumerate the drive maps and then get the free space on each - not exactly easy to locate - and also, the system load on scheduling the script to run frequently enough for your liking is probably going to load the system quite heavily. Your script or tool also needs to run in the 'currently logged in user context' to work.
4. The above won't work on a server or terminal server, where multiple users can be logged in simultaneously.
I'd suggest you might want to start with some powershell, e.g.
Get-PSDrive -PSProvider FileSystem
and then process the results from there.