Constraint handling during tracking operations is at the core of many real-world control implementations and is well understood when dynamic models of the underlying system exist, yet becomes more challenging when data-driven models are used to describe the nonlinear system at hand. We seek to combine the nonlinear modeling capabilities of a wide class of neural networks with the constraint-handling guarantees of model predictive control (MPC) in a rigorous and online computationally tractable framework. The class of networks considered can be captured using Koopman operators, and are integrated into a Koopman-based tracking MPC (KTMPC) for nonlinear systems to track piecewise constant references. The effect of model mismatch between original nonlinear dynamics and its trained Koopman linear model is handled by using a constraint tightening approach in the proposed tracking MPC strategy. By choosing two Lyapunov functions, we prove that solution is recursively feasible and input-to-state stable to a neighborhood of both online and offline optimal reachable steady outputs in the presence of bounded modeling errors under mild assumptions. Finally, we demonstrate the results on a numerical example, before applying the proposed approach to the problem of reference tracking by an autonomous ground vehicle.