There are all kinds of cameras that can be used for web cam shots. As far as I know, all of them require an outdoor enclosure to be used outdoors, and supercircuits.com has some of the lowest prices on those. They also sell cameras of varying prices. Most of the cameras can't operate in temps far below freezing, so they need a heated outdoor enclosure. Netcam (
www.stardot-tech.com) can operate down to -40 degrees, and is one of the best. Axis Communications also makes great stuff.
The kind of camera you want is called a Network Camera - meaning it has a built-in minicomputer so it doesn't need any help sending your images to the web. You CAN use a normal web cam (like the kind that people use for video chat) and those now have great picture and video quality. However, you have to keep it warm and dry, and you have to have it working with a computer that's always on. I know of one cam operator in Michigan who uses such a setup to provide a snowmobile trail cam shot. He built a watertight shack for the computer (trailside) and the camera sits in a PVC electrical junction box that he modified as a camera enclosure. The computer provides enough heat for the camera, which produces a little bit of its own heat.
For help setting up a web cam, google the phrase, "setting up a scenic web cam" or something like that. You'll get plenty of results to sift through, and many of them are private parties who've decided to share their knowledge and experience--some even provide step-by-step pictures they took of their own set-up process. They also share cost-saving tips and how to jury-rig things that would otherwise be expensive.
I often see the WMUR shot of Meredith Bay on the weather reports. I believe their bay-cam is actually a TV camera permanently mounted to the building. As far as I can tell, it's not on their web site, but it would probably be listed under "Sky Cams" in the weather section if it was.