With 10.11(15A243d), Kindle works for the books already in the library, but registering the device (macBook Air) to download new books is not possible.
I haven't been able to download any books I have stored in Amazon Cloud since I first got my Kindle in January.
The iOS app works like a charm, I can download both purchased and uploaded books, but on OS X all I see is Trasure Island, Pride and Prejudice and Aesop's Fables, all of which are there to begin with.
And yeah, now that I decided to re-download the app from the App Store, I chose to deregister to see if that helps with downloading content and alas I'm facing the same problem you Gentlebros have encountered - a blank dialog with no sign-in fields.
This is a non-issue for me since I've gotten used to using the Kindle, iPad and iPhone, but I'd sometimes like the Macbook's bigger screen...
I have same issue, kindle app works fine on IOS 8 and IOS 9 (beta) but with El Capitan, it wouldn't sync, and it refuses to find network connection (tried multiples as well as iphone hotspot) and trying to register it after deregistering it yields a blank page.