The example of the Seatbelts, led by Yoko Kanno comes to mind. All of Cowboy Bebop's soundtrack was composed and performed by them.
Looking at their discography, their albums are all from the Cowboy Bebop franchise, except their live DVD. However, I am unsure as to whether they were created for this purpose, or if they were just a perfect fit.
Are there any bands that dedicate themselves specifically to composing and performing for anime? Or that may be created specifically for a certain series/franchise?