Complex mechanistic computer models often produce multivariate output. Sensitivity analysis can be used to help understand sources of uncertainty in the system. Much of the literature around sensitivity analysis has focused on univariate output, with some recent advances using multivariate correlated output. One promising method for multivariate sensitivity analysis involves decomposition through basis function expansion. However, these methods often require several model runs and may still be computationally intensive for practical purposes. Emulators have been a proven method for reducing computational time for univariate sensitivity analysis, with some recent development for multivariate computer models. We propose the use of generalized additive models and random forests combined with a principal component analysis for emulation for a multivariate sensitivity analysis. We demonstrate our method using a complex agricultural simulator.