Nonholonomic mechanics has received considerable attention in dynamics and control area. However, due to a wide range of fluctuations in the physical world, the ideal mathematical models of mechanical systems with nonholonomic constraints suffer from issues of ignoring the real-world perturbations and physically difficult to realize. Motivated by recent developments in stochastic and constrained mechanics, here we present a stochastic variational formulation for mechanical systems with or without stochastic nonholonomic constraints. We give stochastic variational principles for both stochastically unconstrained and nonholonomic cases under the same framework by deriving the stochastic implicit Hamel equations. Moreover, an interesting example of the stochastic rolling disk is provided to illustrate the proposed method.