Quadcopters have many applications in transportation, agriculture, measurement, construction, disaster management, and so on. To achieve these applications, studies on attaching a mechanical device to the quadcopters is gaining attention. Weight and center of gravity significantly impact the stable flight, and the attachment of the mechanical device can also be detrimental. By considering those issues, we propose a novel, lightweight and soft gripper based on origami technology for grasping operations. The origami gripper utilizes a bistable mechanism, which is a system with two stable equilibrium points. The bistable mechanism is employed to achieve simple gripping control and high-speed actuation. The bistable mechanism is created by self-folding technology using inkjet printing, which provides the paper elastic hinges and rigidity. Complex origami structures can be formed repeatedly without an external energy supply using the self-folding method. We have successfully developed the self-folding origami gripper with the bistable mechanism and mounted it on a quadcopter to grasp objects during flight.