Do you need the base game for Animal Crossing: Happy Home Paradise?
Yes. Animal Crossing: Happy Home Paradise is extra content and only works if Animal Crossing: New Horizons is already on the same account and the same platform. Without the base game you can redeem the code but play nothing.
Same account, same platform
Buy the DLC for the platform where you already own Animal Crossing: New Horizons. Redeeming it on another account or console makes the content unusable, and that cannot be reversed.