I'm sorry for being rather quiet on this, but believe me it's even more frustrating for me. Over the last weeks, I sank way over 40 hours into finding a solution and still came up empty. For me, it's not reproducible when it works and when it doesn't, so their protection against automated access to their web tracking results kind of works.
At this point, I'm pretty much out of options, just like back when Royal Mail made a similar move with the (invisible) captcha. And just as with Royal Mail, accessing the official FedEx tracking API requires you to be the actual shipper of goods (or a solution provider for the actual shipper of the goods requiring both own and their API credentials), which is the reason for the public "rant" the developers of the similarly-named iOS/Mac app published in the article linked by mike.
That said, I will add another FedEx option called "FedEx Website" in the next update, that will just display their website in the app, so you at least have a reliable option to check the results in the app.