There are two that i have found.
#1: Metropolitan Gymnastics in kent (2-3 minutes from south center) It costs 8 bucks for 2 hours of open gym. Open gym is Mon-Fri at 2pm-4pm and then on Saturday from 12-2. The tramp is a flybed. There is also a foampit that is cool to jump into with the spring boards.
#2: Northwest Aerials in Kirkland. It costs 8 dollars your first time and then 20 each time after that (rediculous i know) Their open gym is on Tuesday and Thursday nights from like 7-9 or something. Their tramps are way better than metro's, BUT they are above ground and there is sometimes thier 'team' practicing so you arent always allow on.
I hope that was helpful. If you find another one let me know haha.