Karabiner Elements stopped working after I updated Dropbox because it couldn’t pick up the configuration files that I synced through Mackup via Dropbox.
Dropbox is now fully supported on macOS Monterey by taking advantage of Apple’s File Provider APIs, which means you need to grant additional permission to Karabiner Elements binary files to access the configuration files.
You can fix the issues by granting karabiner_grabber and karabiner_obsserver full disk access. You can find this setting under Security & Privacy in System Preferences.
You can locate these two binary files by using Finder and pressing Command-Shift-G, then you can paste the following path to open the folder:
/Library/Application Support/org.pqrs/Karabiner-Elements/bin/
Move these two files into the Full Disk Access window and restart Karabiner Elements. If you follow the steps above, Karabiner Elements will start working again in macOS Monterey.