When the church began, as we read in Acts 2, we find that God was adding daily those who were being saved. How did people come to be saved? They heard the preaching of the gospel and obeyed it. It has always been the responsibility of the people of God to teach others the truth. This is a responsibility for each Christian (as part of the universal church) and a responsibility for the local church.
Very often individual Christians do their work in conjunction with a local church. A good example is Paul and Barnabas being sent out by the church at Antioch to preach the gospel. The local church at Antioch had a part in their evangelism, but once Paul and Barnabas left on their journey they were acting as individuals, not as a church. (Notice though, that they were not acting as a man-made organization either.)
What is the responsibility of the local church in bringing people to Christ? There are three examples (that I can think of) of how local churches participated in evangelism in the New Testament — supporting evangelists, preaching in the assembly, and preaching outside the assembly.
Supporting evangelists is the easiest way to get involved in this work that God expects from the local church. The church at Philippi sent money to Paul when they could (Philippians 4:14-16). Paul told the Corinthians that it is the responsibility of the local church to share physical things with those who share spiritual things with them.
Do you not know that those who are employed in the temple service get their food from the temple, and those who serve at the altar share in the sacrificial offerings? In the same way, the Lord commanded that those who proclaim the gospel should get their living by the gospel. (1 Corinthians 9:13-14) (ESV)
It is the responsibility then of the local church to supply the needs of the evangelists who work with them. They should also be willing to supply the needs of evangelists who are worthy of support, need support, and are working with others.
The other part of the local church’s work in evangelism is in the preaching and teaching that is done in the assembly.
If, therefore, the whole church comes together and all speak in tongues, and outsiders or unbelievers enter, will they not say that you are out of your minds? But if all prophesy, and an unbeliever or outsider enters, he is convicted by all, he is called to account by all, the secrets of his heart are disclosed, and so, falling on his face, he will worship God and declare that God is really among you. (1 Corinthians 14:23-25) (ESV)
So teaching and preaching that is done in the assembly is not only for edifying the brethren. It is also for the purpose of evangelism — reaching the lost with the gospel. If there is not preaching and teaching that will convict the unbeliever, then there is something missing in the implementation of God’s plan.
In Acts 6, we can see that the work of the apostles in preaching the gospel was in association with the local church in Jerusalem. The church was helping them by supplying men to do other jobs that needed to be done, leaving the apostles free to preach — and much of that preaching was done outside of the assembly.
The church may do evangelism using other methods than just these two examples (such as correspondence courses, tracts, organizing street preaching, etc.). But it is the work of the local church to do the work — not “do the work” through another church or organization.