Soft pneumatic actuators with a corrugated structure have the advantage of low inflation pressure and a large actuation range and have great potential for use in grippers. However, they are generally difficult to have both large deformation and high force density performance. Inspired by the hydraulic joints of spider legs, this paper designs a multi-cavity soft-pneumatic actuator with a corrugated V-fold structure, which realizes bending through the deformation of the air cavity structure instead of the traditional way of squeezing the air cavities against each other, and has significantly large curvature high force density compared with actuators such as PneuNets. At an inflation pressure of 40 kPa, the proposed soft actuator achieves a bending angle of 306° with a curvature of 0.114 rad mm−1, while at an inflation pressure of 45 kPa, the actuator achieves an actuation force of 5.15 N with a force density of 0.0101 mN (mm3.kPa)−1. The modeling approach based on continuum mechanics modeling of soft materials combined with finite element analysis (FEA) (or experiment) is proposed to achieve accurate modeling of the bending angle and actuation force of actuators with complex corrugated structures. In addition, the proposed actuator has made initial explorations into the application of soft grippers and has achieved fairly good gripping results compared to other types of grippers.