"From How to enable or disable Kindle-app screen rotation", CNET, Dec 2013

Step one: Run the app, then load any book.

Step two: Tap the lower right corner of the screen. You should see a little padlock icon.

Step three: While it's still visible, tap that icon to toggle between locked and unlocked settings. When it shows locked, the screen will not rotate from whichever orientation it's currently in. When it shows unlocked, the screen will rotate when you tilt your device.