Whether video or audio, many times, the wireless solution uses as many - or more - wires as the wired solution.

Besides that, "work well and it will be cheap" are oxymorons. Typically, "work well" costs. "Cheap" means different things to different people - and applies to different products, well, differently. For example, a pro-grade, Titan wireless audio/video system is around $3,000. So... what is "cheap"? $1,000? Obviously, for a high school production, something less than that... but you won't find anything for $50 - if that is what you consider "cheap"

But let's explore...

First, a transmitter is needed to wirelessly send the audio and video signal to a base station. The camcorder plus into the transmitter - with a cable (or wire). The transmitter needs a power supply. So it must be battery powered - or connected to power with a cable. So far, two wires... The image and audio are transmitted wirelessly to the base station. The base station connect - in your case - to the projector with a few wires... one for video and one or two for audio... and the base station needs power... even if we count the AV cables as a single "wire", we are up to a total of four wires...

Your solution...
Rather than actually use a camcorder to transmit the video with an AV transmitter, use the camcorder as just a prop. Mount - tape, velcro, whatever - a "wireless security camera" to the camcorder. There are a few that can be powered with a battery.

The least expensive I found, quickly, and from a reputable source is
http://www.homedepot.com/webapp/wcs/stores/servlet/ProductDisplay?storeId=10051&productId=100396419&langId=-1&catalogId=10053&PID=3523498&ci_sku=100396419&ci_src=10043468&cm_mmc=CJ-_-shopzilla-_-D27-_-100396419&AID=10368321&cj=true

In this case, I can't tell if the camera has a built-in mic - but at least you now can look beyond just using the camcorder as the video source - and hopefully, this helps move your production forward...

And if it is a "live news feed thing" and not a studio newscast, the video quality does not need to be that good - most of the Skype-type live field news feeds are not very good video quality at all...

Break a leg!