Divest OS supports Amazon Fire HD 7 and 8 but the installation instructions are primarily for Windows.
TWRP can also be used from linux as explained very well in this video of the LineageOS installation for the FireHD 8 where one can see an internal connection procedure of the device that replaces the fastboot cable.
https://www.youtube.com/watch?v=Iavjnzxnjl4
Personally, I was particularly interested in installing DivestOS on a 2013 Fire HD7 running Android 4.4 which is currently unusable except for reading books.
XDA has a specific tutorial for this model, but it's too complicated for an Android installation that will probably find too few resources to do anything useful, so I'll leave it