The energies at geometries close to the equilibrium for the e(+)LiF and e(+)BeO ground states were computed by means of diffusion Monte Carlo simulations. These results allow us to predict the equilibrium geometries and the vibrational frequencies for these exotic systems, and to discuss their stability with respect to the various dissociation channels. Since the adiabatic positron affinities were found to be smaller than the dissociation energies for both complexes, we propose these two molecules as possible candidates in the challenge to produce and detect stable positron-molecule systems. Moreover, low-energy positron scattering on LiF and BeO targets may show vibrational Feshbach resonances as fingerprints of the existence of stable ground states of e(+)LiF and e(+)BeO.