I think the answer to that would be yes. As long as you pay the deposit with a Disney Visa card, they know you are a cardholder.
I changed a reservation last night (in which the original one had a deposit paid with the Disney Visa) and still haven't gotten the email. It saved us $500, since we had orginally booked the
DDP.
I would just call. If you give them the code, they don't know you didn't get the email. A lot of people are reporting not receiving the email. The deal isn't up on Chase's website yet either.