Prayer is an essential means by which we can commune (fellowship) with God—and not just God as an abstract being, but God as a personal Father, Son, and Holy Spirit. Each member of the Trinity gives himself to us in the work of prayer. Indeed, prayer wouldn’t even be possible if not for the Trinity.
Theologian Carl Trueman writes,
The New Testament makes it quite clear that the human act of prayer is intimately connected to the trinitarian actions of God and is in fact enfolded and subsumed within that larger divine action.
We wouldn’t even pray at all if it were not for the Spirit.
Thus, in Romans 8:26 Paul declares that the Spirit intercedes for believers in their weakness, when they do not know what they should pray for. Even more fundamentally, we wouldn’t even pray at all if it were not for the Spirit. Prayer is a discourse not simply between us as creatures and God as our creator. Prayer is a discourse between us as children and God as Father. And we would not be able to recognize God as our Father if it were not for the Spirit:
For you did not receive the spirit of slavery to fall back into fear, but you have received the Spirit of adoption as sons, by whom we cry, “Abba! Father!” The Spirit himself bears witness with our spirit that we are children of God. (Rom. 8:15–16)
So the Spirit enables us to pray and also brings our weak prayers in a perfected form to the other great Intercessor, the Son of God:
Who is to condemn? Christ Jesus is the one who died—more than that, who was raised—who is at the right hand of God, who indeed is interceding for us. (Rom. 8:34)
The righteous Son has access to the throne of the Father.
In Romans 8 both the Spirit and the Son are called intercessors—they are nonexpendable in the work of our intercession (or prayer). And this by necessity requires the Father, too. To whom else are they interceding, after all? In prayer the Spirit perfects our requests, petitions, and praises and brings them to the Son, who in his authority as the righteous Son of God has access to the throne of the Father, where he makes our prayers his own. This is why we pray “in the name of Jesus”—his name is what grants us access to God. Otherwise we would be shut out on account of our sin and unrighteousness.
Again, Trueman is spot on when he says:
Practically speaking, therefore, a healthy, vibrant prayer life depends to a large extent upon a good understanding of trinitarian doctrine. Only then will we both understand what it is we are doing and have confidence that it will be effective and powerful. A correct doctrine of God as Trinity does not guarantee a healthy prayer life, but a defective doctrine of the Trinity guarantees a prayer life that will be much less than it should be.