I'd start by contacting the third-party and asking them how you can enable and disable routing. They may already have a solution, because ISA can certainly do that. They may require you to call their support desk, though, which isn't much help for you.
You can cut off Internet access with a switch, firewall, proxy server, router, or a pair of electrically-shielded scissors. The real question is, how can you turn it on and off easily from within the classroom? Most firewalls, proxy servers, and many routers, provide a browser interface that you could use to turn routing on-and-off. Even my $100 Linksys home router has this. If you want to be able to turn it on-and-off with the press of a single button, you may need to find a programmer with networking skills, or an administrator with scripting skills.
This was first published in November 2003