Wash. Rev. Code § 35.22.120
On petition of a number (equal to fifteen percent of the total number of votes cast at the last preceding general state election) of qualified voters of any municipality having adopted a charter under the laws of this state, asking the adoption of a specified charter amendment, providing for any matter within the realm of local affairs, or municipal business, the said amendment shall be submitted to the voters at the next regular municipal election, occurring thirty days or more after said petition is filed, and if approved by a majority of the local electors of the municipality voting upon it, such amendment shall become a part of the charter organic law governing such municipality.
RCW 35.22.120
Times for holding elections: RCW 29A.04.311 through 29A.04.330.