We present a new framework for finding the optimal transition paths of metastable stochastic chemical kinetic systems with large system sizes. The optimal transition paths are identified, in terms of reaction advancement coordinates, to be the most probable paths according to large deviation theory for the limiting dynamics governed by stochastic differential equations. Dynamical equations for the optimal transition paths are obtained using the variational principle. A multiscale minimum action method is proposed as a numerical scheme to solve the optimal transition paths. Applications to the toggle switch model are presented.