I would like your thoughts on organizing works in a church that does not have the luxury of elders, such as visitors, attendance, sick members, evangelism. Can men be appointed or volunteer to serve as leaders over these areas with an assistant? Wouldn't this make these areas offices? Do we have authority for this?
Churches are given responsibility for accomplishing various tasks for the Lord. Often you will find them categorized as bringing the gospel to the lost, edifying its membership, and aiding saints in disasters. Those duties exist whether all the offices within the church are filled or not. A church without a preacher still has a duty to teach the gospel. A dedicated preacher makes the effort easier, but the lack of a preacher doesn't mean the work goes undone.
For the same reason, elders are leaders with a congregation, but a lack of elders doesn't mean the work goes undone. Members of a congregation without elders still need to welcome visitors and follow up with them to see if an opportunity exists to teach the gospel. Members need to be aware when someone becomes erratic in their attendance because it might be the first signs of personal struggles with sin. Members need to make sure that fellow members are being looked after when someone is ill. These and other tasks are not dependent on the presence of elders in a congregation.
We know that churches did operate without elders for periods of time. In Acts 13, Paul and Barnabas begin to travel on their first journey to establish churches. On their return journey, probably a year or two later, we read, "So when they had appointed elders in every church, and prayed with fasting, they commended them to the Lord in whom they had believed" (Acts 14:23). The first mention of elders in a church is in Acts 11:30, yet the church in Jerusalem existed for quite a while prior to this time. We don't know when they were selected and the fact that the apostles worked in Jerusalem for much of that time might have eased the need for them. Still, it is interesting that when a dispute arose over a work in the church, the apostles asked the church to appoint faithful men to handle the task (Acts 6:1-6). No mention was made of elders, so we can't conclude if they existed or did not exist at this time, but still there was a task that need to be done and men were appointed to organize this work.
Similarly, no specific mention is made of elders in Antioch though in the same time frame it is repeatedly mentioned that Jerusalem had elders. Yet the church in Antioch (not just the elders if they had any) sent aid into the hands of the elders in Jerusalem (Acts 11:27-30), sent out Paul and Barnabas to work (Acts 13:1-3), heard the report of the accomplishments by Paul and Barnabas (Acts 14:26-28), and sent Paul and Barnabas to Jerusalem to talk to the apostle and elders there about resolving a dispute (Acts 15:2).
We also know beyond the seven men selected by the church in Jerusalem to handle the feeding of widows, that others were appointed to accomplish tasks for the church. Phoebe was doing a service for the church in Cenchrea while at Rome, though we don't know the exact nature of her work (Romans 16:1-2). Mary is mentioned as working hard for the church in Rome (Romans 16:6). Stephanas, Fortunatus, and Achaicus are mentioned as helping Paul on behalf of the church in Corinth (I Corinthians 16:17-18). A brother is mentioned as being appointed to carry relief funds from the churches to Jerusalem (II Corinthians 8:18-22). Thus we know that individuals were appointed to carryout tasks on behalf of the church.
So long as the tasks are authorized, individual members can be given responsibility to carry out those tasks on behalf of the church.