Most webcams are supported either using v4l or v4l2 (Video4Linux). Some require special drivers. I would recommend getting the webcam working on a mainline distro before moving the webcam to TinyCore. Getting a webcam to work could be trivial or crazy depending. Research can be done for Video4Linux support. You'll then need to add v4l(2) support to TinyCore and possibly modprobe some drivers. Hopefully this helps...
If not. Well, sorry... - FRINK