Are carnal believers still co-laborers with God's apostle?

Yes, carnal believers can still be co-laborers with God's apostle. The Corinthian church was carnal, not the most spiritual, yet they were still co-laborers with the Apostle, and this places all believers in the same position—co-laborers together to bring about the eternal will of God.

Les explains that the Corinthians were carnal, not a spiritual church like Ephesus or Philippi, and were beset with divisions—some claimed to follow Paul, Apollos, Peter, or even Christ. Paul had to defend his apostleship because of constant slurs that he was an imposter and renegade Jew. Yet after addressing some problems, he still calls them co-laborers with the Apostle, and Les says that puts us in the same place: all believers are co-laborers together to bring to fruition the eternal will of God.

1 Corinthians 1:9

God is faithful, by whom ye were called unto the fellowship of his Son Jesus Christ our Lord.

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Are believers today co-laborers with the Apostle Paul in God's work?

Yes. The Corinthians, though carnal and beset with divisions, were still co-laborers with the Apostle, and that puts believers today in the same place. All are co-laborers together in God's work to bring to fruition the eternal will of God.