Which system in the frog is responsible for allowing it to move through contracting and relaxing?

It is the muscular system

The muscular system because t he muscular system is the body's network of tissues that controls movement both of the body and within it. Walking, running, jumping: all these actions propelling the body through space are possible only because of the contraction (shortening) and relaxation of muscles. These major movements, however, are not the only ones directed by muscular activity. Muscles make it possible to stand, sit, speak, and blink. Even more, were it not for muscles, blood would not rush through blood vessels, air would not fill lungs, and food would not move through the digestive system. In short, muscles are the machines of the body, allowing it to work.