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Feedback optimization is a control paradigm that enables physical systems to autonomously reach efficient operating points. Its central idea is to interconnect optimization iterations in closed-loop with the physical plant. Since iterative gradient-based methods are extensively used to achieve optimality, feedback optimization controllers typically require the knowledge of the steady-state sensitivity of the plant, which may not be easily accessible in some applications. In contrast, in this paper, we develop a model-free feedback controller for efficient steady-state operation of general dynamical systems. The proposed design consists of updating control inputs via gradient estimates constructed from evaluations of the nonconvex objective at the current input and at the measured output. We study the dynamic interconnection of the proposed iterative controller with a stable nonlinear discrete-time plant. For this setup, we characterize the optimality and stability of the closed-loop behavior as functions of the problem dimension, the number of iterations, and the rate of convergence of the physical plant. To handle general constraints that affect multiple inputs, we enhance the controller with Frank-Wolfe-type updates.