Does the Bible call the local church to the work of cultural transformation?

The short answer is no. The slightly longer answer is that the church is called to the work of cultural transformation only insofar as the church is called to help people grow as disciples of Christ in every facet of their lives.

1. No. The church is called to preach the gospel, disciple Christians, and mark itself off from the world by its membership and discipline (see Matt. 16:18-19, 18:15-20, 28:18-20). This is the church’s mission. This is the church’s responsibility. To devote itself to other matters, however appealing, would be to neglect the commission its Lord gave it.
2. Only as an outworking of discipleship. In the course of discipling Christians, the church should, through its members, have some impact on the culture that surrounds it. This doesn’t mean that the church should devote itself to the work of cultural transformation per se. It does mean that the church should help its members apply the gospel and the Word of God to every area of their lives, which should have some impact for good as members do this in the public sphere.