I think their form is in violation of Visa/Mastercard agreements.
I am not sure about Amex. But for Visa/Mastercard you can not require a form of ID as a requirement for accepting a card.
I am guessing they are trying to cut down on fraud/chargebacks, mainly international.
I would just send the form, but without a copy of the ID or card. But I question how secure their file send page is. They should also remove the instructions telling customers to EMAIL a scanned copy, as that is VERY insecure.
They should be using AVS and VCodes.