Do you want video or just still frames? If it's the latter, then just have it run a simple web server and have a process that grabs a frame from the camera at regular intervals.

You might even be able to figure a way to allow you to tell it to snap a shot when you press a button. Shouldn't be too difficult at all.

This all assumes that it's running a real operating system (which, unfortuantely, may not support the WinTV card; I don't know).