Potential energy doesn't seem to require movement, but it required movement to become potential.
For example an object resting on a table edge has potential energy, PE = m * g * h, where m is the object's mass, g is Earth's gravitational acceleration, and h is height of the table. The object isn't moving, but someone placed it on the edge of the table, and that required expenditure of kinetic energy. KE = (m * v^2) / 2, where m is the object's mass, and v is it's velocity on it's way to the table edge.
The table is dependent on charges moving inside its atoms to keep it from collapsing into the Earth's gravity.
Energy contained in a magnetic field doesn't seem to require movement, but the field is caused by spinning electrons in atoms aligned in magnetic domains. Without the motion of electric charges, there is no magnetic field and no energy.
The binding energy of an atomic nucleus doesn't seem to require movement, but that's because it already has been removed from the nucleus. The mass of a nucleus is always less than the individual masses of the protons and neutrons in it. Nuclear fusion produces gamma rays which carry away the binding energy, and leave the nucleons in a lower energy state than they'd be if separate. That's why they hold together as a nucleus. To separate them again, you need to replace the enormous binding energy they lost.
Energy itself is movement. All particles vibrate. If you have a string theory outlook, the nature of the universe is the vibration of strings. To have energy, you need movement.