Amazon have issued an update to the iOS Kindle app fixing an issues with iOS 7. The update fixes iOS7 upgrade problems where you would have to re-register and re-download your Amazon books after the iOS upgrade.
The app is free from the iTunes Store
What’s New in Version 3.9.2
This is a fix for a Kindle issue with the upcoming iOS7 OS upgrade that may cause customers to have to re-register and re-download books from Amazon. Please download and install – this is a required update before upgrading to iOS7