The Washington State Patrol takes the position the Legislators are protected during session and 15 days before the session from being served civil infractions also known as speeding tickets.

The Washington State Constitution Article II, Section 16 states:

SECTION 16 PRIVILEGES FROM ARREST. Members of the legislature shall be privileged from arrest in all cases except treason, felony and breach of the peace; they shall not be subject to any civil process during the session of the legislature, nor for fifteen days next before the commencement of each session. 

So yes, if you become a legislator you’re immune from speeding tickets by the Washington State patrol 15 days before the legislature starts and during the legislative session.