If someone is "rocking" his or her chair, will it be natural to use:
Hey! Stop swinging on your chair!
Is the use of "swing" natural in this context? It is a normal chair.
The person leans back and moves it (chair) sideways on two legs..
Swinging is a very unusual way to describe this, as FumbleFingers noted in comments. To strictly answer your question, no, it would not be a natural use.
However, if you are trying to emphasize a particularly exaggerated motion or energetic person, and you had the context to support it, you could use swinging in a figurative way as an exaggeration.