"Deliverer" seems like it should be an acceptable alternative to "delivery boy/girl/person", but it isn't. However, this is not because of its phonology, as Omnidisciplinarianist speculates, because "deliverer" is an English word that people use quite regularly in certain circles -- however, it's used to mean something entirely different, so it cannot be used for "delivery person".
While the verb "to deliver" is most often used for bringing someone something and giving it to them, it has several other definitions. One of these definitions is "to rescue, to save, to set free". Consider the phrase "deliver us from evil", from the Lord's Prayer -- it means "save/rescue us from evil". "Deliverer" is based on this meaning of "to deliver" and is a synonym of "savior, rescuer, messiah". You're unlikely to encounter it outside of religious contexts, in my experience, but it's used quite frequently in many translations of the Bible, as well as in sermons, the lyrics of hymns and contemporary Christian music, and in other Christian contexts.
So, is the McDonald's delivery guy your deliverer? Depends how badly you needed that McDonald's.