Well, that didn't last all that long. It seems most people who requested an invite to the new Inbox app from Google are receiving notifications indicating that they're into the cool kids club. As the wave of invites rolled out, the @inboxbygmail Twitter account confirmed that they have "left the building" and to check back soon if you weren't in this wave.
Once you get the invite, you can then download the app from Google Play and it'll automatically know that your account has been selected to work. If you haven't yet requested an invite but still want in, you can always shoot an email to firstname.lastname@example.org — chances now are that you'll be getting an invite sooner than later.