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.