Intercession: the role of the high priest
The letter to the Hebrews was addressed specifically to those members of the early Christian community who were also Hebrews, or descendants of Abraham. Much of this letter draws on Hebrew scripture to explain why this new community does not appoint priests, provide sacrifices, etc.
Priesthood is explored in great detail, and the author continually refers to Jesus as high priest - both in fulfilling the tasks required of the order of Aaron and as an eternal priest in the order of Melchizedek, appointed by God 'not according to a legal requirement concerning bodily descent but by the power of an indestructible life' (Hebrews 7:16).
Melchizedek is considered as 'priest of the Most High God' eternally because this is the only piece of information about him that 'lives on' in scripture:
He is without father or mother or genealogy, and has neither beginning
of days nor end of life, but resembling the Son of God he continues a
priest for ever. (Hebrews 7:3)
The role of 'intercession' is explained earlier in the same letter:
For every high priest chosen from among men is appointed to act on
behalf of men in relation to God, to offer gifts and sacrifices for
sins. (Hebrews 5:1)
The high priest in Hebrew culture was primarily a go-between, a connection between God and man. This task was performed initially by Moses, who relayed the words of God to the Hebrew people and also interceded on their behalf in preventing God's wrath. In some instances it was in the form of 'prayer' (Numbers 11: 1-3), while other times the scriptures describe Moses communicating face to face with God (Numbers 14).
Aaron and his descendants were later appointed priests to perform administrative or symbolic functions of this role such as offering gifts and sacrifices for sins, effectively acting on behalf of those who needed to placate God's anger, and the Levite lineage became law. To the Hebrew people, the priest was their 'source of salvation' - he alone was able to perform the sacrifices through which they were reconciled with God for their sins.
So the author makes a direct link between Jesus' actions and the role of priest:
In the days of his flesh, Jesus offered up prayers and supplications,
with loud cries and tears, to him who was able to save him from death,
and he was heard for his godly fear. Although he was a Son, he learned
obedience through what he suffered; and being made perfect he became
the source of eternal salvation to all who obey him, being designated
by God a high priest after the order of Melchiz′edek. (Hebrews 5:
7-10)
In comparing Jesus' priestly role to that of the Levites and of Melchizedek, the author sets up Christ as eternal priest, whose ultimate sacrifice on the cross serves as an ongoing intercession:
He has no need, like those high priests, to offer sacrifices daily,
first for his own sins and then for those of the people; he did this
once for all when he offered up himself. (Hebrews 7:27)
Conclusion
Christ makes intercession for 'those who draw nearer to God through him', not through prayer or supplications (which could only be performed 'in the days of his flesh'), but 'once and for all' by his perpetual sacrifice on the cross. In the gospel of John, Jesus' words clarify that one's faith in the basis of this eternal intercession (that he directly comes from and returns to God) effectively removes the need for a priest to physically 'act on behalf of men in relation to God':
In that day you will ask in my name. I am not saying that I will ask
the Father on your behalf. No, the Father himself loves you because
you have loved me and have believed that I came from God. (John 16:
25-27)