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In this article, we present data-driven feedback linearization for nonlinear systems with periodic orbits in the zero-dynamics. This scenario is challenging for data-driven control design because the higher order terms of the internal dynamics in the discretization appear as disturbance inputs to the controllable subsystem of the normal form. Our design consists of two parts: a data-driven feedback linearization based controller and a two-part estimator that can reconstruct the unknown nonlinear terms in the normal form of a nonlinear system. We investigate the effects of coupling between the subsystems in the normal form of the closed-loop nonlinear system and conclude that the presence of such coupling prevents asymptotic convergence of the controllable states. We also show that the estimation error in the controllable states scales linearly with the sampling time. Finally, we present a simulation based validation of the proposed data-driven feedback linearization.