Origami sheds light on the design of mechanisms and robots. In this paper, inspired by an origami structure, an enveloping grasping mechanism is proposed. The origami-inspired grasping mechanism (OIGM) is symmetrically designed and has only one degree-of-freedom (DOF), which can be utilized to develop a gripper that can fully enclose the objects. The kinematic characteristics of the mechanism are analyzed. In particular, we integrate a leaf-spring-based bistable mechanism with the OIGM to design a gripper with a passive bistable grasping capability. Triggered by the impact force between the gripper and the object, the gripper can passively grasp objects. To validate the passive grasping capability of the gripper, a prototype is fabricated and the grasping experiment is conducted.