No, making your character a waterbender to be a villain is not mandatory.

Iluq, while she does have her armies, has started to use other means of enforcing her rule and finding the Avatar, such as hiring mercenaries  and outbuying warriors to work for her. So while her army is 95% waterbenders and Water Tribe people, there are people of other nations working for her, or also hunting down the Avatar for their own purposes.