I've read that the order number works, but haven't tried (I recall something about starting with the "W" and dropping the last two digits - yep, just tried minus the last two digits, on the website, and that works). I signed up for UPS My Choice a while back (free), so when I heard that stuff was shipping, I just opened the UPS app and there it was.
(Once you're registered with UPS, they'll alert you to any inbound package to your name/address - you don't need to enter tracking numbers. Pretty useful. FedEx has something similar. Actually, so does USPS - "Informed Delivery".)