Ah, I think I can repro this.
We have a new admin sidebar that’s being rolled out to navigate the admin/moderator areas. If you add the moderator group to admin sidebar enabled groups they have access to Watched Words, but if they’re still on the original ‘top navigation’ style then they don’t have access to the Customize tab (even though it’s visible).
The direct link should still work if you don’t fancy swopping over to the new admin sidebar just yet /admin/customize/watched_words