I'm running the servers and tournaments for a pretty large LAN party coming up in a couple months. We're planning on having a COD4 tournament, however I'm reading that you need the game actually installed and maybe even a valid cd-key, rather than just a dedicated server app.

Is this the case? Am I supposed to have a valid 50 dollar copy of the game to run each of the 3-5 servers on the LAN?

If not, what do I need to do to make this work right?

Thanks for your assistance ahead of time, I'm hoping the game was setup to go easy on us lanners.

All you need is one legit version of the game which you can copy on several machines without any problems concerning CD-keys. At least that's how it has always worked for us on our Lan parties.
Note that if you copy the COD folder, the statistics and unlocks are also copied to the target machines.