We study the energetics and thermodynamics of the initial stages of hydrogen storage by spillover on prototypical metal-organic framework (MOF) and covalent-organic framework (COF) materials. We use density functional theory on periodic frameworks to achieve reliable and accurate predictions for these materials. For pure IRMOF-1, it has been suggested that there is an enthalpy barrier to the addition of the first hydrogen per benzene and that this barrier is removed by hole doping. We do not observe this enthalpy barrier. However, we do observe that the binding energy for the first hydrogen is unfavorable and creates a kinetic barrier without hole doping. Therefore, hole doping by Zn vacancies or other means is still necessary for the hydrogen storage process to proceed. We observe that the optical absorption for the Zn vacancy model is different from that of the pure model; therefore, the presence of Zn vacancies could be experimentally tested. For COF-5, we find that the energy barrier is not resolved by doping. This may explain why it has been difficult to achieve significant hydrogen spillover on COF materials.