I've tried it but no DnD works for me. I guess it's because I don't play those tournament games so I cannot make moves.
In general, it is possible, of course. Either with DHTML or (what I prefer) Java applet. However, no such option works with WebTV, as far as I know. Moreover, although it's quite easy for chess, it could be very difficult for other game types where more than one move is allowed per turn.