In this paper, we focus on the stochastic generalized Nash equilibrium problem (SGNEP) which is an important and widely-used model in many different fields. In this model, subject to certain global resource constraints, a set of self-interested players aim to optimize their local objectives that depend on their own decisions and the decisions of others and are influenced by some random factors. We propose a distributed stochastic generalized Nash equilibrium seeking algorithm in a partial-decision information setting based on the Douglas-Rachford operator splitting scheme, which significantly relaxes assumptions on co-coercivity and contractiveness in the existing literature. The proposed algorithm updates players' local decisions through box-constrained augmented best-response schemes and subsequent projections onto the local feasible sets, which occupy most of the computational workload. The projected stochastic subgradient method is applied to provide approximate solutions to the augmented best-response subproblems for each player. The Robbins-Siegmund theorem is leveraged to establish the main convergence results to a true Nash equilibrium and sufficient conditions to be satisfied by the inexact solver used. Finally, we illustrate the validity of the proposed algorithm through two numerical examples, i.e., a stochastic Nash-Cournot distribution game and a multi-product assembly problem with the two-stage model.