Moir\'e engineering in atomically thin van der Waals heterostructures creates artificial quantum materials with designer properties. We solve the many-body problem of interacting electrons confined to a moir\'e superlattice potential minimum (the moir\'e atom) using a 2D fermionic neural network. We show that strong Coulomb interactions in combination with the anisotropic moir\'e potential lead to striking ``Wigner molecule" charge density distributions observable with scanning tunneling microscopy.