Is there a correct posture for prayer?

The simple answer to this is to come before God in humility. Knowing who God is and having awe and respect Him. We do see from scripture though several examples of how to come to God in prayer:

David gives us one example here:

Bowing: Psalm 5:7 But as for me, by Your abundant lovingkindness I will enter Your house,At Your holy temple I will bow in reverence for You.
On our knees: Solomon gives us an example here: 2 Chronicles 6:13 Now Solomon had made a bronze platform, five cubits long, five cubits wide and three cubits high, and had set it in the midst of the court; and he stood on it, knelt on his knees in the presence of all the assembly of Israel and spread out his hands toward heaven.

We are also told one day everyone will submit on their knees: Philippians 2:10 so that at the name of Jesus every knee will bow, of those who are in heaven and on earth and under the earth,

Lying prostrate: Revelation 1:17 When I saw Him, I fell at His feet like a dead man. And He placed His right hand on me, saying, “Do not be afraid; I am the first and the last,

We also see this from Nehemiah: Nehemiah 8:6 Then Ezra blessed the Lord the great God. And all the people answered, “Amen, Amen!” while lifting up their hands; then they bowed low and worshiped the Lord with their faces to the ground.

and Jesus Himself: Matthew 26:39 And He went a little beyond them, and fell on His face and prayed, saying, “My Father, if it is possible, let this cup pass from Me; yet not as I will, but as You will.”

Lifting hands: Paul gives us an example of this here in 1 Timothy 2:8 Therefore I want the men in every place to pray, lifting up holy hands, without wrath and dissension.

Lifted eyes: This seems a little strange to us because we are used to bowing heads and closing eyes but we see the gazing to the heavens in prayer in the bible.
John 11:41 So they removed the stone. Then Jesus raised His eyes, and said, “Father, I thank You that You have heard Me.

Luke 9:16 Then He took the five loaves and the two fish, and looking up to heaven, He blessed them, and broke them, and kept giving them to the disciples to set before the people.

silence and stillness: Just be with God! Psalm 46:10 Be still, and know that I am God; I will be exalted among the nations, I will be exalted in the earth!

Crying out: Just the opposite of what we just looked at! Sometimes we just let it all out to God. Psalm 55:17 Evening and morning and at noon I will pray, and cry aloud, And He shall hear my voice.

So just remember there are many ways to pray but the main point is humble reverence and awe of God! You can’t go wrong there!

Tom