Yes. It is certainly possible to make a swing oscillate without touching the ground.
An important note (particularly with respect to some of the other answers that have been posted) is that I'm talking about periodic changes in the state of the system, not necessarily motion of the center of mass of the whole system. In other words, a swing-set which is not coupled to the ground (I'll say it is on a frictionless surface, for example) can oscillate, although the frame will move opposite the swinging child such that the CoM of the whole system is stationary. For the CoM to move, the swing-set does indeed need some coupling to the ground, be it frictional, rigid, or other, but in either case there may be oscillations and the system can posses non-zero kinetic and potential energy.
First, from a conservation of energy argument, there is no contradiction. Before even considering the system dynamics and its equation of motion it is obvious that if the child on the swing moves around he/she will influence the state of the swing system. If you are having conceptual issues because no force is being exerted on the ground, and it therefore seems that no work is being done, consider that the child represents a bank of stored energy.
Now, the question is whether the child on the swing can initiate oscillations without touching the ground. Clearly, yes he can. By shifting his CoM, the angle of the swing is forced from its equilibrium, which means the system will now oscillate. For the swing-set on a frictionless surface these oscillations will include the counter-motion of the frame, but they will be oscillations none the less.
Interestingly, once oscillations are initiated, they can be amplified without the child exerting any force along the direction of his motion. Simply by raising and lowering his center of mass along the direction towards the swing's pivot, the oscillations can be amplified via parametric excitation, on which wikipedia has a good page.