Why do you need Windows to access something that's web-based?
We have computer chips in our ID cards, the card reader's drivers only work in Windows. Example: to access my work email in OWA, I have to insert the card into the reader and enter a passcode.
Have you tried plugging the USB card reader into the Mac?
Are you using smart chips? (SIM card looking gold face contacts?)
http://en.wikipedia.org/wiki/Fileifferentsmartcardpadlayouts.jpg#file