We present a theory of coherent shock-induced vibrational excitation of molecules in solids. We treat an idealized impulsive shock wave traveling through a one-dimensional monatomic lattice doped with a diatomic impurity and focus on the ultrafast dynamics occurring within the localized shock front itself. We present a simple classical mechanical model based on the theory of collision-induced translational to vibrational energy transfer, modified to treat the multiple correlated impulsive forces acting on the internal degrees of the molecule as the shock wave passes, and compare the predictions of the theory with classical molecular dynamics simulations. It is found that our approach provides a qualitative description of the behavior observed in molecular dynamics simulations and, in some cases, gives quantitative predictions of the vibrational energy uptake as a function of shock velocity. The potential relevance of the model to shock-induced chemical processes in solids is discussed.