Express Action Sports

What you may be looking for will be in two parts. The camera suitable of shooting at the resolution you need (and maybe even in slo-mo) and a weatherproof housing, designed to take all that abuse.

A camera with those attributes will never be "cheap" or even inexpensive.

Part of your budget for the web show, should include all the tools you need to produce such a program. My guess you will need around $10,000 to capitalize your business even if the labor is free. Post production computer systems while no longer in the $100,000 plus cost range are not based upon a $400 notebook computer with the least expensive video editing program you can find. Figure at least $3,000 for the post production equipment … maybe $4,500 when you include the software and audio hardware
References :
digiPro
filmGrad

You should get a video cam. Consider the well-reviewed, relatively recently announced, Kodak Zx1 Digital Video Camera.
http://gadgetwise.blogs.nytimes.com/2009/01/05/kodak-to-launch-rugged-mini-camcorder/
http://ces.cnet.com/8301-19167_1-10131595-100.html

But if you are one of the billions and billions of buyers out there who think you must have a still camera and video in one (both capabilities won’t be good BTW, one or the other has to sacrifice) then check out the: Pentax
Optio W80, a rugged digital camera featuring a 12.1-megapixel sensor capable of recording 720p video at 30 frames-per-second.

Before you buy contact tech spt at the mfr of your prospective camera choice. Tell them you want to run around with a bunch of kids (or adult males acting like kids) and will be getting paint in & on the camera. Ask if there is a way to clean it off w/out ruining your investment.

When you say your "show" I don’t know how loosely or professionally you’re using that term. So I’ve mentioned more consumer oriented gear. If you are planning commercial grade productions, and can write off equip. as a professional undertaking, then you can clearly go more upscale. Let us know.
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Fisher Price has a digital that might work for you. Cheap, not fancy, but made to stand up to the abuse a little kid would give it.
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Olympus builds a line of "Tough" cameras you might want to check out. Most ‘waterproof’ cameras are also tough rated. You might want to check these out:
http://www.dpreview.com/reviews/compare_post.asp?method=sidebyside&cameras=canon_d10%2Cfuji_finepixz33wp%2Coly_stylustough8000%2Cpanasonic_dmcts1%2Cpentax_optiow60&show=all
http://www.dpreview.com/reviews/Q209waterproofgroup/
http://www.digitalcamera-hq.com/digital-cameras/new-waterproof-cameras_roundup.html
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I think the best option would be this camera:
http://www.amazon.com/dp/B000F5FKHI/?tag=patricksnet-20

It seems to meet all of your requirements: Cheap at $50. Records sound and video. Waterproof, you can even use it underwater. Shock resistant because it is made for action sports.

You could strap it to your bike, helmet, paintball gun, whatever and get first person video really easily. I wouldn’t spend a bunch of money on a fancy camera like the other guy who said it would take 10k to get started. You can buy this one to start out and then upgrade later if you want to.

If you want to spend more you can get extra features with these upgrade models:
http://www.amazon.com/dp/B001A1U4AE/?tag=patricksnet-20
http://www.amazon.com/dp/B001A1O59U/?tag=patricksnet-20
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