Thanks for the great replies. If only it could track itself!
I finally broke down and e-mailed customer support to find out. If anyone is interested, I received the following automated reply (the boldface is my emphasis):
2. Obtaining carrier name and tracking number
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Once the order has shipped, the order status page will give you the name of the carrier and the tracking numbers.
The tracking numbers may not be listed on the site for up to 3 business days after the order has shipped. If the tracking number(s) is listed on the Order Status page, you can click on the tracking number and you will be linked to the carrier's Web site to see the status of the package.
If the tracking number is not listed on the status page, you may be able to obtain the tracking number by calling our automated status line listed above. Often the tracking number will be available on our status line before it is listed on the Web site.
Orders that complete shipping preparation after our shipping cut-off time of 7:00 PM will be picked up by the carrier on the next business day. The shipping timeframe begins on the date the order is picked up by the carrier, not the date the order becomes ready for shipment.
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The good news is that I called the automated number and found out the shipper and tracking number. The bad news is that 3-5 day shipping will be 5 days.

Oh well, I guess that's what I get for ordering so close to Christmas.
Carl