依頼・いらい: To contract with a lawyer, designer, ad firm, etc - any relationship where you'd use the phrase "The client wants. . ... "is いらい。

頼む・たのむ: to ask someone to do something for you. You can use this when ordering food at a restaraunt, but also when asking a favor from a friend. たのむ has the nuance that the person being asked can refuse.

注文:ちゅうもん: to order something at a place of business - usually mail-order. But also ordering food.