I'm not sure how much influence residency forums have (ASCP has one also, although I think it might be called resident council instead). They often are concerned more with educating residents about key issues (like boards prep, finding a job, things like that) than trying to influence major policy. Although I'm sure that varies. Residency forum leaders have a lot of input with CAP leaders though.
I believe every program is entitled to have representatives in the residency forum. From within that group, people can run for the titled positions (president, secretary, etc).