Invites for the official Legion Beta Test (even though what has been presented as an Alpha has been more of a Beta and what we're getting here is a nearly-complete demo of the expansion) are going out.
No, I haven't gotten one, though I'm crossing my fingers that I will soon.
As a PSA, remember that there could be plenty of scam emails floating around sending you to bogus websites. I've historically even gotten scam emails that somehow managed to have a Blizzard email address that proved false when I checked with a GM about the issue at hand.
So just cut the email part of the process out. Instead, just check your Battle.net account on the website or simply go to your Blizzard launcher and look in the Region/Account menu, which should have your active WoW account, a PTR account, and if you are in the Beta, a Beta account.
They're probably going to only invite a handful of people at first to make sure the servers have transitioned correctly over to the Beta state, but I would guess that we'll be seeing larger invite waves later on as they start to stress-test.
Still, Beta or no, we already have a launch date, so while you should absolutely still contribute feedback if you get into the test, it's pretty clear that this is mostly happening to let people get hands on the expansion and get excited about it.