A stochastic proximal-gradient method is proposed and analyzed for solving regularized optimization problems subject to nonlinear equality constraints and nonnegativity bounds. The objective function’s values and gradients are available only through inexact stochastic oracles. The method decomposes each trial step into one component that reduces linearized infeasibility and a second component that seeks optimality while preserving that reduction. Trial steps are accepted or rejected through a relaxed sufficient decrease condition subject to stochastic merit function evaluations. Without assuming unbiasedness of stochastic gradient estimates, it is proved under reasonable assumptions that the method reaches an $\epsilon$-KKT point within $\mathcal{O}(\epsilon^{-2})$ iterations with high probability. This guarantee further implies almost-sure termination in finitely many iterations and an $\mathcal{O}(\epsilon^{-2})$ bound on the expected iteration complexity.