It is perfectly possible to access your Gmail account using Amazon Kindle browser. All you need to do is visit the mobile version at http://m.gmail.com rather than full version. Loading full version will cause some kind of error after you enter your login and password. I don’t think it would render well anyway.
So here you go: you can now send and receive emails using your Amazon Kindle anywhere there is WhisperNet (read Sprint) coverage absolutely free of charge courtesy of Amazon :)