The stochastic vehicle allocation problem addresses the movement of vehicles between locations over a given planning horizon. The demand for vehicles to carry loads between locations is uncertain, and vehicles are assumed able to handle several loads over the course of the planning horizon. Assuming random and coming at random time moments demands, we construct a stochastic model for this transport logistic scheme and derive Gaussian approximation for transport and stock level of goods dynamics. The proposed stochastic model in tandem with stochastic approximation procedure permits to take into account random character of demand for freight services and to supplement the classical deterministic analysis with Gaussian approximation for possible random deviations.