The beach and river left bank area in town is open for camping in designated areas free of charge year round. Scout troops, fishermen, boaters and even an occasional civil war re-enactor make these areas family friendly.
Carson Island (near the paylake and at the trailhead is also camping friendly for a small fee during season.  The Breaks Park has more formal (fee) camping at Ratliff Hole and in the main park in season.
Primitive and remote camping opportunities abound in the area if you're hiking!