Is there anywhere in the bible that requires the formal act of confession? Can't you just always repent personally? It's not like the omniscient deity can't read your mind.
There's nowhere in the Bible that it says everything of the faith is contained in the bible. Though it does say that we should cling to tradition even if not written "Therefore, brothers, stand firm and hold fast to the traditions that you were taught, either by an oral statement or by a letter of ours." 2 Thessilonians 2:15
That being said, the Church usually points to Jesus as having instituted the sacrament of reconciliation here "Whose sins you forgive are forgiven them, and whose sins you retain are retained.” John 20:23