A general method to build the entanglement renormalization (cMERA) for interacting quantum field theories is presented. We improve upon the well-known Gaussian formalism used in free theories through a class of variational non-Gaussian wave functionals for which expectation values of local operators can be efficiently calculated analytically and in a closed form. The method consists of a series of scale-dependent nonlinear canonical transformations on the fields of the theory under consideration. Here, the λϕ4 and the sine-Gordon scalar theories are used to illustrate how nonperturbative effects far beyond the Gaussian approximation are obtained by considering the energy functional and the correlation functions of the theory.