If you are registered with the UPS app, then the package will show up automagically in the app. You don't need to enter the tracking number. I guess it checks your name and address or something. The app also allows you to change the shipping destination to a local UPS partner too if desired.
In the past you could also use the order number in some situations as a kind of reference number, but I can't remember if that was UPS or FedEx. I did try my order number on the UPS website, but I couldn't find the package that way. However, the UPS app had my package listed (even though Apple still also says it's Preparing to Ship).
If you go to the UPS website and select track by reference number. Select Package and put in your billing phone number without any dashes. You should be able to pull your tracking number. That is assuming your iPhone has initiated the shipping process (which is long before apple will tell you it has).