We carry out continuum quantum Monte Carlo calculations of the quantum-mechanical Wigner distribution functions of selected nuclei, up to 16O. These distributions provide a form of quantum tomography of the spatial and momentum structure of the system. They also help identify the location of high-momentum regions in atomic nuclei and provide insight into the onset of alpha clustering. Besides their intrinsic interest, these distributions will be useful for neutrino event generators, as they correlate the positions and momenta of nucleons in the initial target state. To facilitate their application, we address the need to store them compactly by developing an accurate Gaussian Process emulator that automatically preserves their normalization.