Objection 1. It seems that the dispensing of this sacrament does not belong to a priest alone. For Christ's blood belongs to this sacrament no less than His body. But Christ's blood is dispensed by deacons: hence the blessed Lawrence said to the blessed Sixtus (Office of St. Lawrence, Resp. at Matins): "Try whether you have chosen a fit minister, to whom you have entrusted the dispensing of the Lord's blood." Therefore, with equal reason the dispensing of Christ's body does not belong to priests only.
Objection 2. Further, priests are the appointed ministers of the sacraments. But this sacrament is completed in the consecration of the matter, and not in the use, to which the dispensing belongs. Therefore it seems that it does not belong to a priest to dispense the Lord's body.
Objection 3. Further, Dionysius says (Eccl. Hier. iii, iv) that this sacrament, like chrism, has the power of perfecting. But it belongs, not to priests, but to bishops, to sign with the chrism. Therefore likewise, to dispense this sacrament belongs to the bishop and not to the priest.
On the contrary, It is written (De Consecr., dist. 12): "It has come to our knowledge that some priests deliver the Lord's body to a layman or to a woman to carry it to the sick: The synod therefore forbids such presumption to continue; and let the priest himself communicate the sick."
I answer that, The dispensing of Christ's body belongs to the priest for three reasons. First, because, as was said above (1), he consecrates as in the person of Christ. But as Christ consecrated His body at the supper, so also He gave it to others to be partaken of by them. Accordingly, as the consecration of Christ's body belongs to the priest, so likewise does the dispensing belong to him. Secondly, because the priest is the appointed intermediary between God and the people; hence as it belongs to him to offer the people's gifts to God, so it belongs to him to deliver consecrated gifts to the people. Thirdly, because out of reverence towards this sacrament, nothing touches it, but what is consecrated; hence the corporal and the chalice are consecrated, and likewise the priest's hands, for touching this sacrament. Hence it is not lawful for anyone else to touch it except from necessity, for instance, if it were to fall upon the ground, or else in some other case of urgency.
Reply to Objection 1. The deacon, as being nigh to the priestly order, has a certain share in the latter's duties, so that he may dispense the blood; but not the body, except in case of necessity, at the bidding of a bishop or of a priest. First of all, because Christ's blood is contained in a vessel, hence there is no need for it to be touched by the dispenser, as Christ's body is touched. Secondly, because the blood denotes the redemption derived by the people from Christ; hence it is that water is mixed with the blood, which water denotes the people. And because deacons are between priest and people, the dispensing of the blood is in the competency of deacons, rather than the dispensing of the body.
Reply to Objection 2. For the reason given above, it belongs to the same person to dispense and to consecrate this sacrament.
Reply to Objection 3. As the deacon, in a measure, shares in the priest's "power of enlightening" (Eccl. Hier. v), inasmuch as he dispenses the blood. so the priest shares in the "perfective dispensing" (Eccl. Hier. v) of the bishop, inasmuch as he dispenses this sacrament whereby man is perfected in himself by union with Christ. But other perfections whereby a man is perfected in relation to others, are reserved to the bishop.
-Summa Theologica Pt III, Q 82 A3
It seems as though Aquinas didn't approve of Extraordinary Ministers of Holy Communion. Interesting.