We consider optimal control problems governed by linear elliptic partial differential equations (PDEs) with parametric uncertainty and almost sure state constraints. First, we discuss stochastic gradient methods for the numerical computation of solutions. Then we analyze the stochastic semismooth Newton for solving the Moreau--Yosida regularized problems and derive superlinear convergence of the iterates in expectation under suitable sample size growth conditions. For a certain class of PDE constraints, we use monotonicity with respect to the random parameters to construct a deterministic surrogate for the state constraints. The surrogate problem is regularized and path following is adapted to the stochastic setting. Numerical experiments demonstrate fast convergence and effective enforcement of the almost sure state constraints.