The Moon Brothers are among the hill tribes that Tyrion recruits, in return for money and weapons to take over the Vale for themselves.
In the books
The Moon Brothers are perhaps the clansmen who attack Catelyn and her escorts on their way to the Vale.
Ulf and more of the Moon Brothers are among the clansmen who follow Tyrion to the Lannister camp, and subsequently participate the Battle of the Green Fork. Ulf and at least a dozen of his tribe are killed in the battle. Ulf is succeeded by Crawn son of Calor as the clan chief.
In "A Clash of Kings", Crawn and his Moon Brothers serve as guards for Tyrion in King's Landing. Tyrion is careful to keep them away from the Stone Crows.
Prior to the Battle of the Blackwater, Tyrion recognizes that the Moon brothers and the other hill tribes, while ferocious, are undisciplined and inexperienced at siege-warfare, so he sends them south across the river to the Kingswood to harass the scouts and raid the baggage train of Stannis's army as they approached the city. They perform quite well, destroying Stannis's scouts and screening forces to such an extent that he has no warning of the approach of Tywin’s massive reinforcements approaching the city.
The Moon Brothers are not mentioned in the following novels. In the beginning of "A Storm of Swords", Bronn reports Tyrion what became of the Stone Crows, Burned Men and Black Ears (some remained in the Kingswood; some returned home; some were shamefully expelled from King's Landing), but says nothing about the Moon Brothers.