Vibrationally-rotationally inelastic cross sections have been computed for H+SiO collisions over the energy range 200-9600 K. The calculations use a previously calculated ab initio potential energy surface which was fitted to an analytic form [P. Jimeno, M. D. Gray, and G. G. Batint-Kurti, J. Chem. Phys. 111, 4966 (1999)]. Potential optimized discrete variable representation techniques were used to compute vibrational matrix elements of this potential and the infinite order sudden method was then used to compute the required cross sections. The potential is very anisotropic and complicated. This is mirrored in the form of the vibrational matrix elements and in their behavior as a function of the scattering coordinate and of the Jacobi angle. The properties of the potential are markedly different from those of the He+SiO system which has previously been used to model the behavior of both H+SiO and H-2+SiO in interstellar and stellar media. Selected vibrationally elastic and vibrationally inelastic cross sections are presented and discussed. Tables of cross sections covering a wide range of vibrational and rotational quantum numbers, as well as a large energy range, are made available through the journal's EPAPS service. The inelastic cross sections computed here are needed in the modeling of circumstellar SiO maser radiation. Significant structure is observed in the computed inelastic cross sections. This structure is of a complicated nature and might well give rise to the vibrational-rotational quantum state population inversions which in turn leads to the observed circumstellar maser action.