Snapping beams enable rapid geometric transitions through nonlinear instability, offering an efficient means of generating motion in soft robotic systems. In this study, a tendon-driven mechanism consisting of spiral-based metabeams was developed to exploit this principle for producing both reciprocating and non-reciprocating motion. The snapping structures were fabricated using fused deposition modeling with polylactic acid (PLA) and experimentally tested under different boundary conditions to analyze their nonlinear behavior. The results show that the mechanical characteristics, including critical forces and stability, can be tuned solely by adjusting the boundary constraints. The spiral geometry allows large reversible deformation even when made from a relatively stiff material such as PLA, providing a straightforward design concept for controllable snapping behavior. The developed mechanism was further integrated into a swimming robot, where tendon-driven fins exhibited two distinct actuation modes: reciprocating and non-reciprocating motion. The latter enabled efficient propulsion, producing a forward displacement of about 32 mm per 0.4 s cycle ($\approx$ 81 mm/s, equivalent to 0.4 body lengths per second). This study highlights the potential of geometry-driven snapping structures for efficient and programmable actuation in soft robotic systems.