This article proposes a data-driven control framework to regulate an unknown stochastic linear dynamical system to the solution of a stochastic convex optimization problem. Despite the centrality of this problem, most of the available methods critically rely on a precise knowledge of the system dynamics, thus requiring offline system identification. To solve the control problem, we first show that the steady-state gain of the transfer function of a linear system can be computed directly from historical data generated by the open-loop system, thus overcoming the need to first identify the full system dynamics. We leverage this data-driven representation of the steady-state gain to design a controller, which is inspired by stochastic gradient descent methods, to regulate the system to the solution of the prescribed optimization problem. A distinguishing feature of our method is that it does not require any knowledge of the system dynamics or of the possibly time-varying disturbances affecting them (or their distributions). Our technical analysis combines concepts from behavioral system theory, stochastic optimization with decision-dependent distributions, and Lyapunov stability. We illustrate the applicability of the framework in a case study for mobility-on-demand ride service scheduling in Manhattan.