Quantization of a toy model of a pseudointegrable Hamiltonian impact system is introduced, including Einstein-Brillouin-Keller quantization conditions, a verification of Weyl's law, the study of their wave functions, and a study of their energy levels properties. It is demonstrated that the energy level statistics are similar to those of pseudointegrable billiards. Yet, here, the density of wave functions which concentrate on projections of classical level sets to the configuration space does not disappear at large energies, suggesting that there is no equidistribution in the configuration space in the large energy limit; this is shown analytically for some limit symmetric cases and is demonstrated numerically for some nonsymmetric cases.