Hydrogen is a versatile vector for heat and power, mobility, and stationary applications. Steam methane reforming and coal gasification have been, until now, the main technologies for H2 production, and in the shorter term may remain due to the current costs of green H2. To minimize the carbon footprint of these technologies, the capture of CO2 emitted is a priority. The in situ capture of CO2 during the reforming and gasification processes, or even during the syngas upgrade by water–gas shift (WGS) reaction, is especially profitable since it contributes to an additional production of H2. This includes biomass gasification processes, where CO2 capture can also contribute to negative emissions. In the sorption-enhanced processes, the WGS reaction and the CO2 capture occur simultaneously, the selection of suitable CO2 sorbents, i.e., with high activity and stability, being a crucial aspect for their success. This review identifies and describes the solid sorbents with more potential for in situ CO2 capture at high and medium temperatures, i.e., Ca- or alkali-based sorbents, and Mg-based sorbents, respectively. The effects of temperature, steam and pressure on sorbents’ performance and H2 production during the sorption-enhanced processes are discussed, as well as the influence of catalyst–sorbent arrangement, i.e., hybrid/mixed or sequential configuration.