[Fire HD 8 and HD 10 Q&A, Help & Troubleshooting] bucksitter:
Glad I found this thread, as I ran into this exact problem just a day or so ago.

My 2016 Fire HD 8 running Fire OS v5.6.0.0 had just received an update to AquaMail, and I clicked on the icon on the Home Screen to check mail, and received the prompt/message about the SD card not being available (none installed). Also, that icon became grayed-out.

At first I was thrown off and thought it was something due to the AquaMail app, so I went on their support forum and sought help. They didn't have much to offer, especially considering this being a Fire tablet with sideloaded Google Play store. So I sought help from Amazon support.

They were quite good, and via email at first, they suggested I clear the cache and data of the App Store app, restart and check. This had no effect.

During a telephone call (very quick contact initiation) they were able to remote my device using a built-in viewer called Mayday. The support rep looked around at the installed apps and configuration, making sure routine items were in place. The initial rep couldn't suggest anything, and so consulted with someone else, who suggested to Force Stop the Home Pages app/process, then delete cache and data of it, then restart the device. This worked, and the grayed-out AquaMail icon became active again (previous uninstalls/reinstalls of AM had no effect).

I soon discovered this problem again when I was experimenting with the Yahoo Mail app from the Amazon App Store. Same exact symptoms, with the Home Screen icon becoming grayed-out and inactive. Uninstalling the app had no effect on the leftover icon, so I applied the previous troubleshooting fix to this, except I streamlined it to simply clearing the Home Pages cache and Force Stopping it. This allowed the icon to become active again.

I found that when deleting the data from Home Pages that it (understandably) scuttled the order of the apps on the Home Screen, and I wanted to avoid that bit of labor to re-order them, so my routine became clearing its cache and Force Stopping it. Once you back out of all the menus in an attempt to get back to the Home Screen, there is a delay as it restarts itself. Just wait and it will reappear.

I was having subsequent problems with the Yahoo Mail app, where I would do a fresh install and the icon simply didn't show up in the Home Screen. Force stopping

Home Pages (cache clearing optional) always popped it back to visible status. I can't comment on whether this is a property of the latest Fire OS update to 5.6.0.0 or not.

However, with the Yahoo Mail app I'm having other problems, mainly with when I have two Yahoo accounts (one is sbcglobal.net based) installed. Notifications don't reliably work with that account (I have them set up with separate notification sounds, but even if not separate it doesn't notify for the one account. The Home Pages routine/fix doesn't seem to apply for this problem, but I'm thinking there may be another system app/process that may be jinxed like Home Pages that I could fiddle with.

I think the bottom line is that Amazon's customization of Android for v5.6.0.0 was a little off, so they're going to have to take a closer look at this.