We extend the time-dependent Variational Monte Carlo method to continuous-space quantum hamiltonians together with a systematic multi-body expansion of the many-body wave-function. The resulting variational description is applied to the Lieb-Liniger model of interacting one-dimensional bosons. We first show that ground-state properties can be obtained with high precision with ground state energies several orders of magnitude more accurate than previous variational approaches based either on Variational Monte Carlo or continuous Matrix Product States. Then, we study the out-of-equilibrium dynamics induced by a quantum quench in the interaction strength. Our variational Monte Carlo results for the dynamics of the pair correlation at contact are in good agreement with existing exact Bethe ansatz results available for a small number of particles and non-interacting initial states, but also enables the study of large particle numbers and general quench protocols. Analysis of the quench dynamics of a correlated initial state reveals that, far from ``super''-integrable points, the long-term dynamics of local density fluctuations approach closely thermal equilibrium.