Are there any plans or a time table to get the mobile app on the Amazon App Store? My Fire Tablet has too many problems trying to use the app off of Google Play Store.

Yeah!  What he said!

I have it running on my Fire tablet. The underlying system is Android. There's a well know workaround to install Google Play on the Fire. A quick Google search will show you how. It's easy and no risk. Once that's installed the Fire becomes a much better tablet!

But the application directly in the amazon apps store would be better. And since these are android apps it does not have to be so complicated to do it.

I am sure someone will correct me if I am wrong - but I think it has to do with the fact the app is still in the beta stage.  Hopefully, once they have a full release, they will look at moving it onto the amazon app store (they are an amazon company after all) and into the windows store for the mobile variants.

The mobile site works fine on my Fire tablet's browser, but I would like to see an official app for the amazon appstore so that I can download the books to use offline.