The nonlinear model of intramolecular excitations on a ladder lattice integrable by the inverse scattering transform is developed. The model is closely related to the nonlinear Schrodinger model on the same lattice with linear and nonlinear couplings between the chains explicitly taken into account. The pair of auxiliary Lax operators is found and the set of Marchenko-type equations is obtained. The soliton and the reduced soliton solutions of the model are explicitly presented. Even the simplest types of solutions are proved to exhibit both the spatially constricted translational mode typical to the traditional one-chain soliton and the interchain beating mode redistributing the excitations between the chains in a way similar to the linear intramolecular excitations. The possible physical applications of the model are pointed out. The nonlinear model of intramolecular excitations on a multi-leg ladder lattice as well as its continuous counterpart are shown to be integrable too.