Looker's police detective agency exists in a world with apparently no government
Who says there's no government? Just because we don't see them, doesn't mean they don't exist. Nothing in Pokémon, whether it be games, anime, or otherwise, has ever implied that there's no government in the Pokémon World.
And there is confirmed to be a government since there's a government building in Hau'oli city. It'd called the Hau'oli city hall.