You better believe the Church teaches us to pray for conversions. Vatican II declares it itself. From
Apostolicam Actuositatem on the Apostolate of the Laity (please read this entire document):
Here is some more
From
Ad Gentes (On the Mission Activity of the Church) :
Institutes of the contemplative life, by their prayers, sufferings, and works of penance have a very great importance in the conversion of souls, because it is God who sends workers into His harvest when He is asked to do so (cf. Matt. 9:38) God who opens the minds of non - Christians to hear the Gospel (cf. Acts 16:14), and God who fructifies the word of salvation in their hearts (cf. 1 C,or. 3:7).
More from this document (please read the whole document, not just the bits I have posted):
In the present state of affairs, out of which there is arising a new situation for mankind, the Church, being the salt of the earth and the light of the world (cf. Matt. 5:13-14), is more urgently called upon to save and renew every creature, that all things may be restored in Christ and all men may constitute one family in Him and one people of God.
Therefore, this sacred synod, while rendering thanks to God for the excellent results that have been achieved through the whole Church’s great-hearted endeavor, desires to sketch the principles of missionary activity and to rally the forces of all the faithful in order that the people of God, marching along the narrow way of the Cross, may spread everywhere the reign of Christ, Lord and overseer of the ages (cf. Ecc. 36:19), and may prepare the way for his coming.
We are ALWAYS supposed to work towards the evangelization and conversion of others. Here are a few of the many examples of this. I took a few of the many such exhortations from one of John Paul II’s encyclicals. I have also provided other post-concillar encyclicals containing the same message. Please read them all.
“The task of evangelizing all people constitutes the essential mission of the Church … Evangelizing is in fact *the grace and vocation proper to the Church, *her deepest identity. She
exists in order to evangelize”.(69)
“The proclamation of the word of God has
Christian conversion as its aim: a complete and sincere adherence to Christ and his Gospel through faith”.(138)
- The Name of Jesus Christ is the only one by which it has been decreed that we can be saved (cf. Acts 4:12). Because in Africa there are millions who are not yet evangelized, the Church is faced with the necessary and urgent task of proclaiming the Good News to all, and leading those who hear it to Baptism and the Christian life.
John Paul II,
Ecclesia in Africa (On the Church in Africa) September 14, 1995 [Apostolic Exhortation]
Please read these to see the importance of working and praying towards the conversion of all:
Paul VI,
Evangelii Nuntiandi (On Evangelization in the Modern World) December 8, 1975 [Apostolic Exhortation]
John Paul II,
Ut Unum Sint (That they may be one - On Commitment to Ecumenism) May 25, 1995
John Paul II,
Redemptoris Missio (On the Permanent Validity of the Church’s Missionary Mandate) December 7, 1990
John Paul II,
Ecclesia in Asia (On the Church in Asia) November 6, 1999 [Apostolic Exhortation]
John Paul II,
Christifideles Laici (The Vocation and Mission of the Lay Faithful in the Church and in the World) December 30, 1988 [Apostolic Exhortation]