Webcam takes stock cams, builds them up by welding, then grinds them down with new profiles and sends you one. They do NOT grind you a new cam.
So yes, they need your old core, and if you do not send it they charge you up front. Whether or not they'll accept your core and refund your money later is a question you should work out with them, as is the quantity each concerning the valve springs.
Others have dealt with them, but the few times I've dealt with them have been in person, core in hand so I'm not sure what their core refund policy is.