Systems subject to uncertain inputs produce uncertain responses. Uncertainty quantification (UQ) deals with the estimation of statistics of the system response, given a computational model of the system and a probabilistic model of its inputs. In engineering applications it is common to assume that the inputs are mutually independent or coupled by a Gaussian or elliptical dependence structure (copula). In this paper we overcome such limitations by modelling the dependence structure of multivariate inputs as vine copulas. Vine copulas are models of multivariate dependence built from simpler pair-copulas. The vine representation is flexible enough to capture complex dependencies. This paper formalises the framework needed to build vine copula models of multivariate inputs and to combine them with virtually any UQ method. The framework allows for a fully automated, data-driven inference of the probabilistic input model on available input data. The procedure is exemplified on two finite element models of truss structures, both subject to inputs with non-Gaussian dependence structures. For each case, we analyse the moments of the model response (using polynomial chaos expansions), and perform a structural reliability analysis to calculate the probability of failure of the system (using the first order reliability method and importance sampling). Reference solutions are obtained by Monte Carlo simulation. The results show that, while the Gaussian assumption yields biased statistics, the vine copula representation achieves significantly more precise estimates, even when its structure needs to be fully inferred from a limited amount of observations.