I don't think they can do this. They are trying to set up a procedural roadblock that has it all backwards. The applicable law is RCW 9.41.300(2)(a), which reads:
(2) Cities, towns, counties, and other municipalities may enact laws and ordinances:
(a) Restricting the discharge of firearms in any portion of their respective jurisdictions where there is a reasonable likelihood that humans, domestic animals, or property will be jeopardized. Such laws and ordinances shall not abridge the right of the individual guaranteed by Article I, section 24 of the state Constitution to bear arms in defense of self or others;
I don't believe this provision allows them to effectively say "In all of Thurston County there is a reasonable likelihood that humans, domestic animals, or property will be jeopardized so it's all off limits unless an individual can prove otherwise", which is what they are trying to do. They should not be able to relieve themselves of their burden to establish the likelihood of such jeopardy before enacting such restrictions.