What is the proper term for performing the sacrament of the Anointing of the Sick on someone?
The Catechism of the Catholic Church states the following:
1516 Only priests (bishops and presbyters) are ministers of the Anointing of the Sick. It is the duty of pastors to instruct the faithful on the benefits of this sacrament. The faithful should encourage the sick to call for a priest to receive this sacrament. The sick should prepare themselves to receive it with good dispositions, assisted by their pastor and the whole ecclesial community, which is invited to surround the sick in a special way through their prayers and fraternal attention.
Thus bishops and priests are the ministers of the Sacrament of the Anointing of the Sick. And as such they are the ones to administer this sacrament.
The proper phrase you are looking for is: Administering the Sacrament of the Anointing of the Sick.
The Oil of the Sick
The oil used in administering the sacrament of Anointing of the Sick is called Oil of the Sick. It is one of the three Holy oils blessed by the bishop of the diocese at his cathedral on Holy Thursday morning, the other two Holy Oils being Holy Chrism and the Oil of Catechumens, which is used in Baptism.
Oil of the Sick is pure olive oil—nothing being added except the blessing of the bishop. Its appropriateness as part of the outward sign of Anointing of the Sick is evident from the healing and strengthening effects which are characteristic of olive oil.
The essence of the sacrament lies in the actual anointing and the short prayer which accompanies the anointing.
In giving the sacrament, the priest anoints the sick person on the forehead and hands. During this anointing, the priest says: “Through this holy anointing may the Lord in his love and mercy help you with the grace of the Holy Spirit. May the Lord who frees you from sin save you and raise you up.” - The Anointing of the Sick: Comfort and Healing
Canon Law states:
§1. Every priest and a priest alone validly administers the anointing of the sick.
§2. All priests to whom the care of souls has been entrusted have the duty and right of administering the anointing of the sick for the faithful entrusted to their pastoral office. For a reasonable cause, any other priest can administer this sacrament with at least the presumed consent of the priest mentioned above.
§3. Any priest is permitted to carry blessed oil with him so that he is able to administer the sacrament of the anointing of the sick in a case of necessity.