place an order for an item or service

(いらい) To hire a lawyer, designer, ad firm, etc. Put another way:any relationship where you’d use the phrase “The client wants. . … ” is いらい。
(たのむ) To ask someone to do something for you. You can use this when ordering food at a restaurant, but also when asking a favor from a friend. たのむ has the nuance that the person being asked can refuse.   $$$
(ちゅうもん) To order something at a place of business – usually mail-order. But also ordering food. Also, if you ask your friend to do an annoying and detailed favor, like, “Could you do the laundry again with this other kind of soap!”, that’s 注文.