In this research, we address the construction of random effective constitutive equations by nonlinear stochastic homogenization of hyperelastic heterogeneous materials for which a high number of random parameters is needed to characterize the uncertainties at the microscopic scale. We base our new approach on a non-concurrent multiscale method recently proposed for computing the homogenization of nonlinear heterogeneous materials which still represents a difficult task. This technique, based on a numerical construction of the strain energy density function associated with a microstructure, allows this difficulty to be encountered in a deterministic framework. However, in order to obtain an efficient mechanical model, one must take into account the different sources of uncertainties related to the material at the microscopic level.