In this paper, we propose a data-driven control framework for locally stabilizing unstable periodic orbits of discrete-time nonlinear systems. Specifically, we explore the scenarios where the locations of the orbits are not precisely known. In our framework, we use a Pyragas-type delayed feedback controller. This controller uses the difference between the current state and a delayed version of the state as feedback to the system. We show that the system under our controller can be described by another nonlinear system with a particular structure. The periodic orbit stabilization problem for the original system is then characterized as an equilibrium stabilization problem for the new system. For this new system, we investigate local exponential stabilization while paying special attention to situations where neither the location of the equilibrium nor the linearized dynamics around that equilibrium are precisely known. To handle such cases, we develop a data-driven framework that accounts for the scenarios where the difference between the state and the equilibrium is not observable. In our framework, we design the gain of a stabilizing controller by using the data generated through a nonlinear projection of the state.