Abstract: Variational inequality problems allow for capturing an expansive class of problems, including convex optimization problems, convex Nash games and economic equilibrium problems, amongst others. Yet in most practical settings, such problems are complicated by uncertainty. To contend with precisely such a challenge, we consider a stochastic generalization of the variational inequality problem and its extensions in which the components of the mapping contain expectations. When the associated sets are unbounded, ascertaining existence requires having access to analytical forms of the expectations. Naturally, in practical settings, such expressions are often difficult to derive, severely limiting the applicability of such an approach. Consequently, our goal lies in developing techniques that obviate the need for integration. Instead, our emphasis lies in developing sufficiency conditions for claiming existence that are required to hold in an almost-sure sense. We begin by presenting almost-sure sufficiency conditions for stochastic variational inequality problems with single-valued and multi-valued mappings. Next, we extend these statements to quasi-variational regimes. Finally, we refine the obtained results to accommodate stochastic complementarity problems. The applicability of our results is demonstrated on application instances drawn from nonsmooth Nash games and strategic behavior in power markets.