Basically you'll just have to locate the driver for the webcam in the kernel sources (if it's a kernel-space driver). After that, it's a matter of building it, moving the driver onto Chumby, loading it into the kernel, and setting up some program to deal with capturing the images from the camera.
Logitech's cameras are reasonably well supported by the OSS community, but to really find any more information about it, you'll have to google for it.
Displaying the images locally can be done just by dumping the image into the framebuffer, or you can do it a bit smarter by making a Flash widget view the Chumby's own webserver, and serving the images from there. Something to keep in mind: Chumby doesn't have much room for persistent storage, so you need to be smart about how you're storing your webcam images, if you're going to be doing it locally and not on an NFS server. Also, if your camera uses Video4Linux, you're probably going to have to rebuild the kernel (someone else could back me up here, I haven't delved through the kernel .config yet).