A - Thanks for the question! The short answer is no. The reason why can be found in the Bible and the Catechism of the Catholic Church.
First, the bible. In James 5: 13-15:
Is anyone among you suffering? He should pray. Is anyone in good spirits? He should sing praise. Is anyone among you sick? He should summon the presbyters of the church, and they should pray over him and anoint (him) with oil in the name of the Lord, and the prayer of faith will save the sick person, and the Lord will raise him up. If he has committed any sins, he will be forgiven.Notice that when someone needs healing they should call on the presbyters (presbyter = priest), and the anointing of the priest will lead to healing and forgiveness. The two parts of this Sacrament are healing of body/mind and forgiveness of sin. Therefore, because the commission was only given to priests, only they can preside in the Sacrament.
The Catechism states in paragraph 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.