I think you can - on the 4 folks (out of 15) in my office running various incarnations of Windows, I've deleted the IE shortcut from their desktops (Windows complains a lot about this, but it can be done with no harm whatsoever - you can either drag it to the trash, right-click and select delete/move to trash, and in some older versions of Windows the ability to not display it is buried in either the advanced desktop or IE setting). Installed FireFox and let it become the default browser, and put a Firefox shortcut in the same spot as the big blue E resided and a second shortcut in the quicklaunch bar next to the start menu, also deleting the IE shortcut in the quicklaunch bar.
This way, the only time IE will run is when using Windows Update.
Anyhow, I feel your pain - the computers that are the most trouble in our office are the Windows machines...