Starting with macOS Mojave (10.14), Apple has put in new security measures when a program attempts to control another program on the system. Zotero's Word for Mac plugin requires the permission for Zotero.app to be able to control Word. When you first interact with the Zotero Word plugin after updating to macOS Mojave, you'll get a prompt asking for this permission:

If you press “Don't allow”, Zotero won't be able to provide the Word plugin functionality and every subsequent attempt to use the plugin will trigger the “Missing Permissions” prompt, until you follow the steps in the prompt:

1. Open System Preferences
2. Click on “Security & Privacy”
3. Select the “Privacy” tab
4. Find and select “Automation” on the left
5. Check the checkbox for “Microsoft Word” under “Zotero”
6. Restart Word

If you're running Word 2011, be sure you've updated to the latest version, 14.7.7, for compatibility with Mojave's new permissions system.