I don't recall being asked for a Visa card when I registered with Amazon's affiliate program. And I don't see any credit card information in my Amazon Associates account settings. But it looks like I used the same email address to register as I use on my Amazon account that I buy stuff with, and that account does have a Visa card attached to it, so I can't be certain whether or not my Amazon Associates account is linked to my Visa card.
But I can't imagine why they would need a Visa card -- it's an affiliate program; they pay you, not the other way around. What they need in order to pay you is your preferred payment method, and your bank account info (if you choose to be paid via direct deposit).
Are you sure you're attempting to sign up for the right program? You're not on the page to register as an Amazon seller?