Ask the admin to install firefox. Typically admins are quite friendly folk. Be aware that skirting rules by finding ways to run exe files will likely draw the ire of the admin and if they discover it, they will be less likely to install firefox for you than if you asked up-front.
Other than that, you could try a java-based web browser such as Lobo which is a “.jar” file.
Failing that, you could perform the requests manually using telnet from cmd. (And get uber nerd-points). But that’s not really a browser.
