Amazon has released a software update for the current-generation Colorsoft and Paperwhite that lets you double-tap on the sides or back of your Kindle to turn the page.
The new feature is listed in the release notes for the 5.18.1 update and can be found in Settings > Device Options.
The feature works well, requiring two taps in rapid succession, and is not easily triggered by mistake.
It is limited to either jumping to the next page or scrolling down a list of books, and it is unclear if other Kindle models will receive the update.