Hi Jeremy,
Hopefully I can help you figure out if your reservation went through the system or not. First, if you were logged in to My Disney Experience, it will typically use a stored credit card for your reservation, so it might not ask you for one directly (I believe it just asks you to confirm using the card already on your account), so that could be a reason why the reservations system didn't ask you for your credit card number.
Second, the dining reservations system also should have sent you an email detailing your confirmation number, party members, time of reservation and location of your reservation. Double check your email and see if you have something in your inbox; you can enter your confirmation number into My Disney Experience to link it to your account if it did not do that automatically.
Finally, if you did not receive an email, I would suggest
speaking to a Cast Member on the Dining Reservations line; they are incredibly friendly and if, for some reason, your reservation was not completed, they will be more than happy to assist you in completing your reservation.
I hope this helps you out!
Best Wishes,
Amy B.