When I saw that Kindle didn't work after I installed Yosemite I just figure I would use my iPad to download my Kindle books and wait for an update from Amazon. I waited, and waited for an update and there was no announcement or link telling me that new software was available. It would just crash every time I opened up the app.
I was hopeful when OS X 10.10.1 was released but that didn't fix the problem either. Needless to say, I wasn't happy. So what's a voracious reader to do? In the words from the IT Crowd "Have you tried turning it off and on again?"
Yosemite places all incompatible applications in a folder called, funny enough, "Incompatible Software". This folder is located at the root level of your hard drive. Open that folder and delete the Kindle application.
Once that's done just open up the Apple App store and re-download Kindle.
The version on my computer didn't change at all. It still showed up as 1.10.6. But once that's done the application will work just fine. Pass this PSA to all your reading friends. They'll probably thank you.