Woot isn’t Amazon. They don’t have the same resources as Amazon. Besides, the Woot web site works great via a mobile browser so it’s not like Fire users are totally out of luck.
Dev support for all kinds of apps has never been much of a priority for Amazon Appstore users, if they even have a version on Amazon’s store at all… Yes, it seems perfectly logical for Woot to support Amazon’s hardware but if Amazon stops supporting it, then it’s also logical that Woot would stop, too.
I got one of the WTF dollar Fire tablets knowing it’s an older device and that it wouldn’t be supported. I made that choice. I have other tablets of various brands that stopped receiving OS updates and are nothing more than paperweights now. Have you looked into whether you can flash the device with a custom ROM? I used to do that with an HP Touchpad until it became pointless to bother. If the hardware is still good, then there might be unofficial support for it out there.