Objection 1. It would seem that even a layman can confer this sacrament. For this sacrament derives its efficacy from prayer, as James declares (James 5:15). But a layman's prayer is sometimes as acceptable to God as a priest's. Therefore he can confer this sacrament.
Objection 2. Further, we read of certain fathers in Egypt that they sent the oil to the sick, and that these were healed. It is also related of the Blessed Genevieve that she anointed the sick with oil. Therefore this sacrament can be conferred even by lay people.
On the contrary, Remission of sins is given in this sacrament. But laymen have not the power to forgive sins. Therefore, etc.
I answer that, According to Dionysius (Eccl. Hier. v) there are some who exercise hierarchical actions, and some who are recipients only. Hence laymen are officially incompetent to dispense any sacrament: and that they can baptize in cases of necessity, is due to the Divine dispensation, in order that no one may be deprived of spiritual regeneration.
Reply to Objection 1. This prayer is not said by the priest in his own person, for since sometimes he is in sin, he would not in that case be heard. But it is said in the person of the whole Church, in whose person he can pray as a public official, whereas a layman cannot, for he is a private individual.
Reply to Objection 2. These unctions were not sacramental. It was due to the devotion of the recipients of the unction, and to the merits of those who anointed them that they procured the effects of bodily health, through the "grace of healing" (1 Corinthians 12:9) but not through sacramental grace.
Source: Summa Theologica (New Advent)