In Isaiah 22, we see the transferal of the office of the vice regency of the House of David from Shebna to Eliakim. Such is the nature of an office.
Thus, there is no reason to believe that the office of the vice-regency of the eternal House of David passed out of existence with the death of Peter, and, in fact, historically, the office of the vice-regency of the eternal House of David was passed on from Peter to Linus continuously to this day through to Pope Benedict XVI.
There is also evidence of Apostolic succession in Acts.
Acts 1:12-26Then they returned to Jerusalem from the hill called the Mount of Olives, a Sabbath day's walk from the city. When they arrived, they went upstairs to the room where they were staying. Those present were Peter, John, James and Andrew; Philip and Thomas, Bartholomew and Matthew; James son of Alphaeus and Simon the Zealot, and Judas son of James. They all joined together constantly in prayer, along with the women and Mary the mother of Jesus, and with his brothers.
In those days Peter stood up among the believers (a group numbering about a hundred and twenty) and said, "Brothers, the Scripture had to be fulfilled which the Holy Spirit spoke long ago through the mouth of David concerning Judas, who served as guide for those who arrested Jesus— he was one of our number and shared in this ministry."
(With the reward he got for his wickedness, Judas bought a field; there he fell headlong, his body burst open and all his intestines spilled out. Everyone in Jerusalem heard about this, so they called that field in their language Akeldama, that is, Field of Blood.)
"For," said Peter, "it is written in the book of Psalms, " 'May his place be deserted; let there be no one to dwell in it,' and, " 'May another take his place of leadership.'Therefore it is necessary to choose one of the men who have been with us the whole time the Lord Jesus went in and out among us, beginning from John's baptism to the time when Jesus was taken up from us. For one of these must become a witness with us of his resurrection."So they proposed two men: Joseph called Barsabbas (also known as Justus) and Matthias. Then they prayed, "Lord, you know everyone's heart. Show us which of these two you have chosen to take over this apostolic ministry, which Judas left to go where he belongs." Then they cast lots, and the lot fell to Matthias; so he was added to the eleven apostles.