The Hedge-Father is a jolly sort of Hedgehog. He travels the world on a certain winter night to deliver presents to all the good Hedgehogs of the world. Unlike other holiday figures, the Hedge Father does not imprison animals and force them to fly him about. Instead, the Hedge Father simply rolls himself into a ball and teleports himself from home to home.
You don't have to be a member of the Shrine to receive favours from the Hedge Father, you merely need to truly believe in him. So, go ahead and tell the little guy what you want, you may just get it!