I am not expert historian, but I would challenge the idea that your church can possibly be Reformed without elders. First, you need to look no further than the 1689 Baptist Confession of Faith and all Presbyterian churches since the 1600’s to know that the mainstream Reformed tradition always had elders.
But beyond that, what is the connection between eldership and Reformed Theology? This is just a theory, really. But I see at least three connections. One, Reformed believers are those who believe that the Bible is our sole final authority in matters of faith and practice. So when it comes to how to structure a church, the biblical model is a plurality of elders leading the church. There might be other polities that “work.” But that does not change the fact that God has spoken. And since the Word is our final authority, it seems Reformed churches usually end up seeing that eldership is the way to go.
Secondly, biblical church leadership is one step removed from the right preaching of the Word and the right administration of the ordinances. Reformed theologians have taught us that a true church is one marked by the right preaching of the Word and the right administration of the ordinances (Romans 10.13-17; Romans 6.1-4; 1 Corinthians 11.23-26). God has designed that multiple qualified men oversee the preaching and the ordinances. A preaching elder needs the accountability of other qualified men to keep his preaching in order. Though I would not argue that a plurality of elders is essential to the local church the way the gospel, baptism, and the Lord’s Supper are, I would argue that the elders, in God’s design, help to protect those things.
Third, and most important, eldership helps people understand the gospel better. Reformed Christians were “gospel-centered” before that became a popular phrase. The Reformation helped the Church remember that all of Scripture is a testimony to Christ and His saving work. Elders uniquely point people to Christ in word and deed.
Eldership is actually an idea that develops throughout the Bible: Shepherds are a frequent picture of humble leadership. Abel was a shepherd. David was a shepherd. Jesus is the Good Shepherd. Elders are called to shepherd the flock. And in the NT, the idea of eldership is not directly associated with age, as it appeared to be in ancient Israel, but rather spiritual maturity (1 Timothy 3.6). Elders help people to see that Jesus is creating a spiritual people with His blood.
And the unity that is found in a healthy council of elders is a great picture of gospel unity. The church as a whole is always meant to grow in gospel unity, but that is a requirement for an elder board. Perhaps no other group on earth—maybe not even a husband and wife—work to have the kind of unity that a healthy eldership works toward. When functioning properly—though not perfectly—elders in the local church help people to know Jesus better.