In this study, we propose a soft pneumatic gripper that uses a tendon-driven soft origami pump. The gripper consists of three pneumatic soft actuators that are controlled by a tendon-driven origami pump. An external air compressor that supplies air to the pneumatic actuator is replaced by an origami pump. The soft actuator is composed of silicone (Ecoflex 00-30) with a chamber-based structure, which is fabricated using a mold, and the origami pump is fabricated by folding a Kresling patterned polypropylene film. In addition, we conduct a series of experiments to evaluate the performance of the pneumatic actuator with a tendon-driven origami pump. Specifically, movement characteristics, frequency response, blocking force, and the relation between bending angle and pressure are analyzed from the results of the experiments. Furthermore, we understand the entire operation mechanism from the deformation of the origami pump to bending through pressure. Finally, we demonstrate the grasping of objects with diverse shapes and materials, and indicate the feasibility of the pneumatic gripper as an independent module without an external compressor.