MacOS – this globe icon with blue down arrow in the menu bar

I'm fairly sure that's an agent for LANDesk.

From this support page:

This starts at startup with the little three-chevron logo in the menu bar

I had this exact same issue, and user Tom Gewecke's advice seems to have fixed it for me. I simply removed two plist files in ~/Library/Preferences, and then rebooted. The files were com.apple.CharacterPicker.plist and com.apple.CharacterPaletteIM.plist. I don't know which of those files had the issue, but after removing them and rebooting they've regenerated, and everything's good again.

If you're comfortable with Terminal

You probably know how to move files around, just move these two somewhere else, or rename them:

1. ~/Library/Preferences/com.apple.CharacterPaletteIM.plist
2. ~/Library/Preferences/com.apple.CharacterPicker.plist

If you just want something you can copy and paste, here you go:

mv ~/Library/Preferences/com.apple.CharacterPaletteIM.plist{,-backup}
mv ~/Library/Preferences/com.apple.CharacterPicker.plist{,-backup}

Reboot after this, and fingers crossed, it will work for you again.

If you're not so comfortable with Terminal

Open up a Finder window and press Shift ⌘ G (Shift Command G), then type ~/Library/Preferences/, and press Enter. There will be many files in that folder, you're looking for the ones called com.apple.CharacterPaletteIM.plist and com.apple.CharacterPicker.plist. When you find them, you can probably just remove them, but it's always a good idea to back them up. Just move them to another folder or something.

When you've done that, reboot your computer, and see if the Special Characters window works again.