There is most certainly an internal centripetal force within the material keeping it in circular motion. In real objects, this is provided by the intermolecular/interatomic forces, which are never fully rigid, which. This is why the Earth has an equatorial bulge arising from its rotation. Without these forces holding everything in place, the object will cease to exist as a single object. In idealized rigid bodies, the distance between any two points remains constant (which is what it means to be a rigid body), so the internal forces will be whatever is needed to maintain the constant distances, much like forces of constraint.
Since the internal forces are internal, they do not apply any external torque. Angular momentum will be conserved regardless of whether these forces are present.