If you book the room through the airline you cannot get the dining plan. The dining plan can only be purchased as a PACKAGE which includes room and tickets unless you are military, DVC, or AP holders.
I don't believe you will be able to add the dining plan onto these reservations. Adding the dining plan requires that you book a package together including the room/tix/dp... you can't it on as a separate component. Never booked through an airline personally, but other 3rd party bookings work the same way.
You can purchase your tickets online, in person, or over the phone.
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