The whole issue of evangelism begins in the OT with the premise that we are created in the image of God. While this means more than just this, it definitely means we are made for relationship with God. The kind of relationship that is one of love, best understood perhaps by the parent-child analogy; in an ideal sense. Then this relationship was ruptured through the fall of Adam and Eve that brought corruption and fragmentation. Our relationship with God was marred and we need redemption.
From this point on humanity needed saving. The whole story of the OT is the first part of God's great plan to restore that broken relationship built on the call of Abraham, the establishment of the people of Israel as his people, and the up's and down's of their relationship with God based on the Sinai covenant. The call of Abraham concerned not only his family, but the whole of the world, all nations blessed through him.
This suggests from the beginning that mission is utterly central to the faith. God is seeking relationship with every living person in all nations, an eternal relationship of love and shalom.
Hence, we are to do everything we can to ensure that this occurs. God's business is our business.
It is interesting to reflect on the issue of the people of Israel's relationship with God. Their story is one of ups and downs, falling in and out of relationship with God; of faithfulness and idolatry. Take the Judges cycle for example; Israel would be delivered by a judge, live in peace for a period, fall back into idolatry and sin, experience trouble at the hands of enemies, and another judge would rise up and declare the call of God on the people and drive out the enemies. In the monarchy it was the prophets who continually called Israel back declaring God's word to Israel.
The prophets to me prefigure the ministry of evangelism. They called the people of God to their foundational event of salvation (the Exodus) and called people back into relationship with God. They warned Israel of the consequences of their sin (exile, enemies, material improvision etc). They promised blessings for them if they repented.
Their foundational means of calling people was through the divine dabar, the word of God. It was the word that brought creation to bear ex nihilo. It was the divine dabar that proclaimed the coming salvation through the Exodus from Egypt. It was the divine dabar that established the covenant on Mt Sinai. It was the word of God that called Israel again and again back to God. It was the divine word of God that declared the hope of a Messiah who would be God's agent of salvation. And not just to Israel, but his light to the nations, taking God's salvation and shalom to all peoples that the original intention of God for humanity and his world can be restored.
Hence, the proclamation of the word and human salvation in terms of relationship with God lie at the heart of the history of God's dealings with his people. It is no surprise then that as we turn to the ministry of Christ, the apostles and evangelists that this notion of the divine word (dabar, logos) carries over; God achieves his purposes through the declaration of his word by this authorised agents. So the OT suggests, yes 'we' have to declare the divine dabar of God so that all humanity across all nations will hear the divine word of God and potentially be saved. It remains to look through the NT to work this through.