Note that it can be ANY credit card ... The machine is just trying to extract your name, not the account number or anything.
The checkin kiosks will also take the barcode of any Delta document in your itinerary. This would include a boarding pass from a previous segment in your itinerary, for instance.
So check-in online, print out your boarding pass, then you've got what you need to get the machine to work at the airport.
Note that even IF
you check in online, you STILL have to use the machine (or a counter) to check luggage.
It's also handy to see if any better seats have opened up day-of-flight (which often happens) and you can change seats too.
Lastly, if you are so inclined, you can purchase an upgrade right at the kiosk ...