In this study we look into the treatment of parametric uncertainties and the response of a vehicle system with several degrees of freedom, using the polynomial chaos approach. The research aims at investigating the accuracy of the method compared with traditional Monte Carlo simulations, and the change in computational efficiency as the number of uncertain parameters and stochastic external excitations increase. A 7 degree-of-freedom full-car dynamic model has been developed. Key parameters of the system have been assumed to be stochastic, with large uncertainties. In addition, the vehicle runs over rough (and undeformable) terrain, with uncertain terrain height. The polynomial chaos expansion and the Galerkin approach are used to quantify the uncertainties and to determine the time evolution of the stochastic system under sole or combined sources of uncertainties.