Yeah, that would be awesome!
Kindle

There's an app for all versions(I presume) of the Kindle Fire. I wouldn't reccommend it though. Personally I think it sucks; I don't like the interface, the lack of disabling chat in Live, the clocks sometimes get stuck on a specific time...
There are Kindle tablets (with Android, so runs lots of software) and Kindle readers (which don't have Android). Which are you talking about?

Oh, I think he(she?) has the Kindle reader. There's no app for Chess.com from a reader, only tablets.

Has Chess.com put out an update for chess.com ap for Kindle Fire. Mine requests an update and it only lets you see the old ap and it wont let you sign in. Its throws you into a loop where it asks for the update you click ok. Then it takes you to the Kindle Ap store you click the ap icon and it takes you right back to the game where it repeats the same update request. I have uninstalled and reinstalled ap. etc...HELP! I love to play chess on my kindle untill now @ Chess.com

I have the same problem - Kindle app goes into a loop and won't update. I've even removed the app and reinstalled it from the store.
For some reason the app in the google play store has been updated, but the amazon app store doenst seem to know that there is a new version out.
You therefore need to run the installation process from the google play store on the kindle fire.
Its a bit of a pain, but this is what you need to do:
1) Uninstall chess.com from the kindle fire
2) Go here: http://apps.evozi.com/apk-downloader/
3) Copy and paste the following text in, and click generate (no quotes): "https://play.google.com/store/apps/details?id=com.chess&hl=en"
4) Click the "click here to download chess.com" link ( should download a 4.8mb apk file)
5) Get this file onto your kindle (I emailed it to myself and downloaded, but connecting the kindle to your computer should work just fine)
6) Invoke the apk file. I navigated to the file in AndroXplorer, but there are probably many other ways.
If all goes according to plan, you should see the chess.com install process.
Note: if you have never installed an apk manually before, you probably need to got to settings -> device -> Allow Installations of Applications and set it to true.
The Kindle is a disaster, please don't tell me. Still, an app for the Kindle from Chess.com could help.