We report elastic differential, integral, and momentum transfer cross-sections for H2X molecules (X: O, S, Se, and Te) obtained at the static exchange level of approximation. The energy range considered was from 2 up to 30 eV for H2O and from 5 up to 30 eV for the other molecules. Our calculations were performed with the Schwinger multichannel method with pseudopotentials [M. H. F. Bettega, L. G. Ferreira, and M. A. P. Lima, Phys. Rev. A 47, 1111 (1993)], combined with a Born closure procedure in order to account for the long-range potential due to the permanent dipole moment of the targets. Our calculated cross-sections for H2O and H2S are in good agreement with other theoretical results. Agreement with available experimental data is also encouraging. It was found that molecular size plays a crucial role in the scattering process. The influence of heavy and H atoms in the collisions is also discussed. For the integral cross-sections of the heavier molecules we also investigated incident energies below 5 eV, looking for possible shape resonances. Through the symmetry decomposition of the integral cross-sections and the eigenphase sum analysis, we found shape resonances for H2S, H2Se, and H2Te at the B-2 symmetry. For H2Te, we have also found a shape resonance at the A(2) symmetry. For all molecules a very broad structure was found at the A(1) symmetry. This is the first work to report such resonances for H2Se and H2Te.