At launch, Kindle Cloud reader was only available for Chrome and Safari. Today, Firefox users can utilize.
From an Amazon.com Announcement:
Today, Amazon announced that Kindle Cloud Reader, the HTML5-based web app that lets customers read their Kindle books in their web browser, is now available for Mozilla Firefox so the hundreds of millions of Firefox users can start reading their Kindle books instantly, simply by opening their web browser. To start reading, go to http://read.amazon.com using Chrome, Safari on iPad, Safari on desktop and now Mozilla Firefox.
Kindle Cloud Reader is available for Safari on iPad, Safari on desktop, Chrome and, starting today, Mozilla Firefox 6 and above. Kindle Cloud Reader on the iPad is optimized for the size and unique touch interface of iPad. Without even leaving the app, customers can start shopping in the Kindle Store and will find a unique and immersive shopping experience built specifically for iPad’s Safari browser.
Direct to Kindle Cloud Reader