I have some good news and some bad news for you today.
The good news is that Amazon has released an update for the Fire tablet that, in the words of one user, allows the Fire tablet to monitor compatible home cameras with Alexa.
The bad news is that the update also introduced a bug. Numerous Reddit users are reporting that the update has made it difficult to impossible to sideload apps.
For those not familiar with the term, sideloading is the name for when an Android user downloads and installs an app from a source other than an official app store. You might download the app from the developer's own site, or from deals sites like Humble Bundle, and then copy it to your Android device before installing it.
This is a common practice, but thanks to this bug, it no longer works on some Fire tablet models. Instead, users are reporting that the "install" button is grayed out, and cannot be clicked.
And yes, this is very likely a bug; for one thing, Amazon isn't letting everyone download it (they probably pulled the update so they can fix the problem), and here is the official word from customer service:
Thank you. I have replicated the issue and it appears to be a bug with the software update. I have forwarded this information to the technical team along with the details and they will be looking into this issue and working on this. However, any outcome regarding the issue will be in the form of a software update.
You can wait until Amazon sends out fix, but if you need to sideload an app in the meantime, there various ways to sidestep the bug.
One user reported that they plugged in a USB mouse to their Fire tablet, and used that to click the install button. Another said that they disabled a background process (Whisperplay daemon) and eventually they could install apps again.
Going from the install screen to the home screen and then back to the install screen also reportedly bypassed the bug. And finally, pressing the power button to put your Fire tablet to sleep in the middle of the install, and then waking it up, fixed the problem for at least one user.
I have not been able to confirm any of these fixes because Amazon won't let me install the update.
But I do hope they work for you.