You don't actually have to have the login details to track through Amazon. All you need is the link in the order status that leads to the tracking info. You can get full tracking info from the link without logging in and without having the actual tracking number.
It does require an additional click to view the full details, but all the info is in the HTML source without logging in and without the extra click.
Perhaps the app developer has just not gone through the process of parsing the source. Amazon seems to present Chrome with easier to parse HTML than it does to IE 11. I found the info in both browsers without being logged in.