As Christ followers we ultimately do not follow the Law of Moses as though we were national Israelites, however, we do still see principles clearly of God's will displayed throughout the Old Testament. This is so even of the Torah portion of the Old Testament, the first five books.
In the heart of those books, you find much mention of prayer in Leviticus. In fact, Leviticus is essentially a guide to prayer in the appointed place of worship in the nation of Israel, the Tabernacle Tent of Meeting and later the Temple. For them, this was a place prescribed for worship, instead of building one's own altar, as people had done previously. Prayers still could be done elsewhere, but the Tabernacle/Temple became the main and required place for Israelites to properly approach the Lord at certain festivals and high holy days of Israel.
So in Leviticus, perhaps we can echo godly believers then, Leviticus chapter sixteen was the high holy day in the Israelite calendar. It was a prayer meeting, as the people fasted and prayed, and then the high priest did offerings and also essentially prayers (you can find some online of what they likely prayed) before the Lord. The day was even a foreshadowing of our awesome Savior Jesus Christ, who is our substitutionary atonement for sin before God.
A website online that many use, states there are no Bible verses on prayer in Leviticus. This could not be further from the truth. The heart of the the use of the word forgiveness and also dealing with the reality of sin is in early and mid Leviticus chapters. Leviticus is said to be the Romans equivalent of the Old Testament. We find much on prayer.
Chapters 1-3, 4-5, and 16, and 20 have much to say about prescribed prayer and then things not to pray/do related to prayer. In fact, properly understood, these chapters teach even Christians about God's holiness, He is totally set apart and worthy, and God's forgiveness, and God's needing to be reverenced in prayer. It is hard to imagine an Old Testament let alone a New Testament theology apart from the holiness and honor given to God in such passages of prayer.
In fact, our book of Hebrews in the New Testament deals heavily with Leviticus' doctrine. It is the necessary prequel to understand it's wonderful New Covenant theology.
In short, a solid study of prayer in Leviticus exists, and while maybe not the most exhaustive study you or I would ever do, will definitely help you to carry out effective prayer because Leviticus points to who God is and the necessity of Jesus Christ, the perfect spotless Lamb of God for our atonement.