in principle, a web site cannot access any client side hardware, for obvious security reason... And that is why a web only approach to application development is a bad idea in my opinion.
so the ONLY way this could be done is if the card reader manufacturer has created some kind of sdk that goes around these restrictions in order to give access to a web page.
That of course will be specific to a brand/model of reader, if it exists, and will be available through the manufacturer only.