How would you get the ball rolling?
Contact Seattle Parks Foundation - tell them you would like to organize a campaign to raise funds to install the type of equipment you're describing. They will be able to give you a little bit of administrative support, and help you identify the proper city officials to get this going. (And possibly tell you if there is already another similar project underway.)
Fundraising shouldn't be too hard - I would happily shell out 25 bucks towards a fitness trail reasonably close to my house - I am sure there are many others who would do the same.
Based on the current budget situation, though, I think it is going to take someone (who could be YOU!) putting in a lot of hours with no compensation to drive the project forward.