I was having the same error with the OneNote app from Amazon store. I also have OneDrive installed and it was and is working flawlessly. I installed Outlook mail with no issue and it is working correctly.
With OneNote, I would login in and after or during sign in, I would get:
"There's a problem
We can't complete this task. Try again.
Error code: 80004005"
I tried signing in with the wrong password and got an error about wrong username or password. It doesn't seem like a sign in error, but after sign in when the application goes to sync notebooks.
My solution that worked for me:
I uninstalled OneNote from my Kindle Fire 10 (2017 Model)
I installed 1MobileMarket from www.1Mobile.com. Had to go to the site and download an .apk file.
Opened and installed the .apk file.
To do so I had to allow "installation from unknown sources."
Directions at this link - https://www.amazon.com/gp/help/customer/display.html?nodeId=201482620
I opened 1Mobile Market and installed their version of OneNote.
It works great as a native application. Even the clipper "OneNote Badge" works.
It is a step above using the web version through the silk browser.
Hope this helps some folks with the issue.
It seems to like the error and issue is with the Amazon Store version of OneNote.