Find out who your facilities manager is and tell them you need a replacement. Tell them it is bad for your back or whine about ergonomics and they will usually cave in.
Ours are replaced only if complained about. However ours are those fancy Herman Miller chairs that are 800 bucks a chair. No way our company will buy everyone new chairs on a regular basis.
Or be a douche bag and wait for carpet cleaning day. Then stay late, find someone's chair that is in better shape than yours and swap the chairs. Then if you are caught blame the carpet cleaning crew for accidentally putting the chairs in the wrong cubes/office.