Fabric and textile materials are widely used in many industrial applications, especially in automotive, aviation, and consumer goods. Currently, there is a lack of automatic solutions for rapid and effective fabric handling operations that can be expanded to various applications, causing economic loss, workplace safety issues, and process bottlenecks. As a bio-inspired novel technology, soft robotic grippers provide new opportunities for the automation of fabric handling tasks. In this research, an elastomer-based tendon-actuated soft gripper for fabric pick and place tasks is developed through a model-based design approach. Based on finite element analysis, the gripper design is simulated, modified, and validated. Multiple design variables and their impacts are studied. Detailed motion patterns of the underactuated structure are obtained. After the design is established, a prototype is fabricated trough additive manufacturing and overmolding processes to physically test the functionality of the gripper and further validate the simulation results.