You might want to think about a portable karaoke machine. There also may be some "boom boxes" which accept microphone inputs, in which case you'd want to buy a halfway-decent mic. Be aware, though, that running either at top volume will eat batteries. It's just the nature of the beast. Might want to try rechargeables if the unit sees enough use.

Thanks, steve o. Can you recommend any specific brands or models?

Here's a system I found by googling, but it's not very expensive. I thought we would need to spend at least $300 or $400 dollars, but if something less expensive would work for us, that would be great. We can continue to use the borrowed cd player. Is this what I should be looking for?

Here is a link that might be useful: AudioTechnica ATW202 Freeway Handheld Wireless System

The Audio-Technica looks decent. You'll have to feed its signal into whatever is amplifying the CD player. Then my question is if whatever that amplifier is could have both the CD player and the microphone "live" at the same time (assuming that's something you want to do). You also need to consider how you're going to amplify and voice the microphone sound for rugby games (since everything would need to be battery- or generator-powered).

I am not very familiar with karaoke machines, but I've linked to search results for "karaoke" at a company that I've purchased from several times over the years. Those listings might answer some questions for you (and raise some others).

Here is a link that might be useful: J&R listings for karaoke