We consider the homogenization of random integral functionals which are possibly unbounded, that is, the domain of the integrand is not the whole space and may depend on the space-variable. In the vectorial case, we develop a complete stochastic homogenization theory for nonconvex unbounded functionals with convex growth of generalized Orlicz-type, under a standard set of assumptions in the field, in particular a coercivity condition of order $p^->1$, and an upper bound of order $p^+