Is the Washington State Supreme Court required to provide a lock box?

"In addition, the local legislative authority shall provide either a stationary locked box sufficient in size for pistols and key to a weapon owner for weapon storage, or shall designate an official to receive weapons for safekeeping, during the owner's visit to restricted areas of the building. The locked box or designated official shall be located within the same building used in connection with court proceedings. The local legislative authority shall be liable for any negligence causing damage to or loss of a weapon either placed in a locked box or left with an official during the owner's visit to restricted areas of the building."

The Supreme Court isn't in conjunction with any "local legislative authority". It's in conjunction with the state legislative authority.

Are they exempt from having to provide a lock box?

This was a discussion topic some time ago and I believe the consensus was that they don't have to.

I may be wrong on this but logic says that the Supreme Court is part of the Legislative Branch and not subject to regulation by the Legislature. The are responsible for interpreting the law and it's application but are allowed to make their own regulations and rules as to how they conduct their business.

Subordinate courts probably have some oversight by the legislature.

Again, I might be wrong on this but I don't see the State "Supremes" as having to do so.

Supreme Court is Judicial, but on the level of the state, not the local governments, and the state legislature was silent on lock boxes for the Supreme Court.

As I said earlier, they probably don't. Since they also travel, and hear arguments in local jurisdictions, while there the storage issue is clear. The Local Court has to provide.