There is definitely nothing that prevents another applet (JavaCard, not PKCS#11 -- that's just a generic interface) from running on smartcards currently running the EMV applet. Or any other smartcard from running more than one applet -- this is a result of ISO7816 more than EMV.
But my point is that if EMV had chosen to use X.509 instead of inventing their own certificate format then we could re-use the existing infrastructure with zero cost.
Passing arbitrary APDUs from webpages to your card isn't a great idea and it will be SLOW since it will require multiple round-trips -- it's already slow enough just reading objects from smartcards. This is the mechanism used by Citrix/MS Remote Desktop, which is to forward a PC/SC connection -- this has a disadvantage that the remote side must have drivers and is also real slow, but the advantage that the resource can be used for anything, just like it were local.
For what it's worth, I'm the author of some smartcard middleware (CACKey) and have worked with the EMV specification and EMV applets to a lesser degree.