From what you've told me, it seems as if you've got several different people working on the machine - that could be part of the issue, relating to what user rights each person has. It is possible, that someone with admin rights, went into the local security policy settings and established software restriction policies, thinking that this would be a way to prevent specific users from "browsing the net" while working. If this is the case, which I doubt anyone would admit to if you were to ask them, you need to remove the restrictions set in that policy.
Do the following, just to check the settings:
Click the START button, go to Control Panel (if the background is blue, and there are only a few choices, click "Switch To Classic View" at the top left of the window), Double-Click Administrative Tools, Double-click "Local Security Policy", when that window opens, you'll see a list of things on your left. The second-to-last one are the Software Restriction Policies. All you need to do is click once on the "folder" in front of the words, and look in the right pane. If it says "No Software Restriction Policies Defined", then there are a few other things you should try. I don't know how many characters I've typed, but there is a limit - so all I will say from here is that if there are no restrictions set, GET FIREFOX. :)
That will solve all your problems. *ducks from the keyboard the Microsoft guys over ---> there just threw at me* ;)
