For anyone who wanders in off of google or wherever:
Just started running FAF a day or two ago, and I was experiencing a similar issue. The issue was that C:\ProgramData (the default folder FAF wants to use for its data) is unwriteable, and while I'm sure with enough finagling you could 'convince' your computer to let you have permissions for the folder...

But alternatively, and TL;DR - Move your FAF Client Data folder. Click on the menu (top-left corner, the three horizontal lines), select Settings, a pop-up window appears. Click on the tab that says Data and caches. The top item is Data Location, defaulting to C:\Program Files[something or other]. Make a new folder somewhere else, preferably as far away from default Windows directories as possible.

I've experienced similar issues with programs and apps not working right when they use Windows-created directories, so I suspected this was the issue, and sure enough it worked for me. The whole permissions thing probably would work, but I tried it (I'm not exactly an IT guy, but I do know my backslashes from my forward slashes) and Windows 10 is just so ridiculously painful about back-end account management that I was just like "whatever, so much easier to move to a different directory".