We present derivation and implementation of the Multi-Configurational Strong-Field Approximation with Gaussian nuclear Wave Packets (MC-SFA-GWP) -- a version of the molecular strong-field approximation which treats all electronic and nuclear degrees of freedom, including their correlations, quantum-mechanically. The technique allows, for the first time, realistic simulation of high-harmonic emission in polyatomic molecules without invoking reduced-dimensionality models for the nuclear motion or the electronic structure. We use MC-SFA-GWP to model isotope effects in high-harmonics generation (HHG) spectroscopy of methane. The HHG emission in this molecule transiently involves strongly vibronically-coupled $^2F_2$ electronic state of the $rm CH_4^+$ cation. We show that the isotopic HHG ratio in methane contains signatures of: a) field-free vibronic dynamics at the conical intersection (CI); b) resonant features in the recombination cross-sections; c) laser-driven bound-state dynamics; as well as d) the well-known short-time Gaussian decay of the emission. We assign the intrinsic vibronic feature (a) to a relatively long-lived ($ge4$ fs) vibronic wave packet of the singly-excited $ u_4$ ($t_2$) and $ u_2$ ($e$) vibrational modes, strongly coupled to the components of the $^2F_2$ electronic state. We demonstrate that these physical effects differ in their dependence on the wavelength, intensity, and duration of the driving pulse, allowing them to be disentangled. We thus show that HHG spectroscopy provides a versatile tool for exploring both conical intersections and resonant features in photorecombination matrix elements in the regime not easily accessible with other techniques.
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