I have never ever seen an online payment processor that was capable of using a card reader to perform a transaction from a webpage (on a non-specialized device). I don't think there is even any established standard for using a smartcard from a website. WebUSB/WebNFC may work (although browsers have blacklists of vendor IDs to disallow access to e.g. Yubikeys, so at least some smartcards may not be accessible this way), but that's all experimental and questionable stuff.

It might've been possible someone had something like that in ol' good '00s with ActiveX, but that must've been surely an exception (and a security nightmare).