Today (January 3), I called customer service, which has no record of the trace that was started on December 26. They told me _exactly_ what they told me on December 26 (I'm going to submit a form to my supervisor we need to trace the card to get the number blah blah bleeping blah, we can't do anything without the number).
Not believing this is any different from my previous experience with delia's customer service, I'm going to do two things:
(1) call customer service daily until they reissue the card to the recipient
(2) post this, and hope that maybe somewhere over there someone will decide I'm enough of a bad PR problem that maybe they should reissue the card to the recipient.
Whatever else may be the case, it's clear that their internal systems don't keep track of numbers on physical cards ordered online and sent through the mail, so if one goes missing before it gets to the recipient, you are out of luck -- the money is _gone_. At a minimum, you'll have to expend a lot of time and energy to extract it out of customer service.
And that's assuming this works at all.
ETA: Since I purchased it using PayPal, I am pursuing a claim through the Resolution Center simultaneously. I expect to Learn from this experience which resolution process (phone seller, claim through PayPal, whine on blog) is most effective.