Is there a current release of LO that will work on Kindle Fire tablets? If so, where can I find it?
As far as I know fire-tablets are runinng with an custom android where you can not install other software out of the box.
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So first steps would be to get the possibility to install other software or replacing the OS with android, if available.
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Even then there is no full LibreOffice for iOS or Android. You may be able to use the derived version of Collabora:
https://www.libreoffice.org/download/android-and-ios/