This paper proposes an online adaptive control framework for tracking Optimal Periodic Control (OPC) trajectories of feedback linearizable plants with unknown parameters. The literature already offers different online solution algorithms for solving OPC problems. However, the existing algorithms either assume perfectly known plant models or allow for the appearance of unknown parameters in the plant model, but rely on the plant's open-loop stability for implementation of the optimal solution. This paper, in contrast, develops an adaptive feedback linearizing algorithm to simultaneously estimate and track the optimal OPC orbit. Using Lyapunov analysis, the paper shows that the system trajectories always remain bounded and asymptotically approach the periodic solution corresponding to an estimate of the unknown plant parameter vector. Furthermore, when the regressor vector of the parameter estimation law is persistently exciting, global convergence to the true periodic solution trajectory is guaranteed. The paper concludes with a numerical implementation of its adaptive optimal control algorithm for a periodic drug delivery application, commonly used as a benchmark in the OPC literature.