The question comes down to the simple statement: if it uses ActiveX, then the answer is "no it will never work because Microsoft is actively keeping ActiveX proprietary". If they are just using the .NET routines to do the backend work and are outputting html then there is a good chance that it will "just work" (and you can also try FireFox as sometimes their JavaScript is a bit closer to IE's). ASP.NET and Web Forms both produce code that isn't too bad by default, and so they should be useable.
There is also the possibilty that they are creating .NET applications and just triggering them from web pages (usually a bad idea), and in that case there would be a very small chance that those applications would run in .Mono (and thus become cross-platform).
If half your district is using Mac's and the district IT produced something that they won't work with, then your district should re-evaluate the IT department.