This paper proposes a new robotic platform based on origami robots and reconfigurable modular robots. The concept combines the advantages of both robot types into a mobile, quasi-two-dimensional, lattice-type reconfigurable modular origami robot, Mori. A detailed description and analysis of the concept is validated by the presentation of a first prototype that incorporates the key functionalities of the proposed system. The modular robot prototype is mobile, can be connected to other modules of its kind, and fold up to create task-specific three-dimensional reconfigurable structures. Three implementations using the prototype in different configurations are presented in form of individual modules, modular reconfigurable surfaces, and applied to closed-loop object manipulation. The experiments highlight the capabilities and advantages of the system with respect to modularity, origami-folding, mobility, and versatility.