This is not necessarily an answer, but an additional possibility/question to the assembled horde... Is it possible to set up a site that needs a user name, password and their Visa number to get access? Yeah, it's a pain in the ass to remember a Visa number, but that's a piece of information that people wouldn't share. Maybe show them a mask of their Visa number and have random blank spots...
Example: xxxx-xxxx-____-xxxx or ____-xxxx-xxxx-xxxx
That way they don't have to input the entire number, but they do need to have the card handy. Or memorize their number.
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