Can a Catholic priest adopt a child?

Assuming civil law allows the clergy to adopt, an answer would have to address whether Catholic canon law permits it. It would seem not because fathering children would detract from his obligations to the Church (which is a reason why priests cannot marry); he certainly could run an orphanage for such children.

According to this, it isn't a situation covered by Canon Law. I couldn't find any relevant canons in the 1917 Code (or commentary) or the 1983 Code (or commentary).

If the Pope could, then a priest could.

It is commonly believed that Pope Pius IX "adopted" a boy in very extenuating circumstances in 1856 and created in international scandal.

The Mortara Affair began on this date in 1856 when a contingent of papal police kidnapped 6-year-old Edgardo Mortara from his parent’s apartment in Bologna because Church higher-ups had learned that the child had been secretly baptized by a maid five years earlier when he was seriously ill. Edgardo was whisked to Rome, “adopted” by Pope Pius IX, and eventually became a priest. Jewish organizations and international leaders protested to the Vatican and called for Edgardo to be returned to his parents, but the pope declared that canon law forbade non-Christians in the Papal States to raise a Christian child, even if the child was their own. Protests in France led to the founding of the Alliance Israelite Universelle in 1860, and helped fuel the movement for Italian unification, which overcame the Papal States in 1870. As an adult, Edgardo was sent on missionizing missions to Jews in Germany and the United States, and vigorously sought the beatification of Pope Pius IX — which finally took place in 2000. Mortara did reconnect to his parents and attended his mother’s funeral in Bologna in 1895. He himself died at 88 in 1940. - June 24: The Mortara Affair

Fr. Mortara with his mother and a brother.

Here is what Wikipedia has to say of the Moratara Case.

Pope Pius IX's determination to keep Edgardo developed into a strong paternal attachment. According to Edgardo's memoirs, the pontiff regularly spent time with him and played with him; the Pope would amuse the child by hiding him under his cassock and calling out: "Where's the boy?" At one of their meetings, Pope Pius told Edgardo: "My son, you have cost me dearly, and I have suffered a great deal because of you." He then said to others present: "Both the powerful and the powerless tried to steal this boy from me, and accused me of being barbarous and pitiless. They cried for his parents, but they failed to recognize that I, too, am his father.

As Brasshat points out in his answer that it is possible for a priest to adopt a Child. I know of some American nuns (sisters) who have receive permission to adopt children.

Blessed Charles de Foucauld (1858-1915) began a program of buying slaves in 1902 in order to free them. For Blessed Charles this was done as a form to encourage adoption. In addition to being adopted himself, he is also one of the Patron Saints of the Adopted in some Catholic circles.

Our Father never failed to add: «Someone will say to me: “But what about me, I have no family to speak of, I have no traditions, I am one of those serfs produced by modern society; so what is to become of me?” I shall answer: “Well, allow yourself to be adopted by Father de Foucauld!” Here we have a family which is open to us through the generosity of this heart of gold. Just as he loved all the members of his own family, he would have desired to have brothers and sisters, and even an entire people of lay folk who would be attached to his work, that is to say, who would share his love for neighbour and then for those furthest away. So, if we have no family, let us allow ourselves to be inserted into this great family of age old France and of the Catholic Church!» - BLESSED CHARLES DE FOUCAULD

Yes, a Roman Catholic Priest can adopt children. In 1981, the Vatican granted permission to Fr. George Clements, of Holy Angels Catholic Church in Chicago the permission to adopt a child, and he later adopted three more. I expect another Catholic Priest seeking to do so would need permission, but not necessarily from the Vatican.