An extended Enskog asymptotic expansion of the Boltzmann equation is used to adapt the transport theory formalism to polyatomic gas mixtures in strong nonequilibrium situations in which the vibrational degrees of freedom are strongly coupled to the chemical kinetics. The new method allows the explicit inclusion of the vibrational nonequilibrium distributions in the calculation of the transport coefficients. We apply this new formulation to the calculation of the heat conductivity in the flow of binary mixtures of N-N₂ and O-O₂ in a supersonic nozzle.