Free-electron quantum optics provides a versatile platform for manipulating electrons at the quantum level with potential applications in quantum information technologies. We propose a grating-based architecture for fully quantized free-electron quantum optics, in which photon-electron interactions map onto Jaynes-Cummings and Tavis-Cummings models via Bloch-Floquet analysis. Within this framework, we design ultrafast single- and two-qubit gates with cavity-free flying electrons, enabling universal quantum computing in experimentally accessible setups. More broadly, this framework establishes a platform for probing free-electron quantum optics and advancing quantum technologies in simulation, sensing, and information processing.