Oh dear, you may be in trouble. Delta is right you do have to deal with Orbitz. I would call them and see what they can do for you. The problem is Delta didn't get paid by you they got paid by Orbitz. It's a little confusing, but that's the way it works. Most online TAs won't deal with getting you the credit because they have no system really set up to track the credit etc.
(Just an FYI for other shoppers, the "on line" TAs tend to be very difficult to work with in these instances. If you are buying on the net, always see if you can get the same fare are the airline directly to avoid the "middleman". )