Because we travel from Nevada to Washington state and back sometimes several times during the winter in the motorhome (and toad) for the past decade or so, I've chatted through the years with several state troopers from both Washington and Oregon and in both states, it is required especially for heavier vehicles to carry chains if going over the designated roadways during the months from late fall to early spring whether the road is snow covered or not.
Yes, in both states, a trooper will not
pull you over specifically or exclusively to check that you are carrying chains but they can check if you are involved in an accident (even when it's not your fault, i.e. somebody slides into you on an icy road) or other situation where you might need assistance from an LEO (i.e. skidding off an icy roadway). By law, they can cite you for not carrying chains in the proper size to fit your vehicle. It's up to the discretion of the trooper as to whether he wants to cite you for not having them or having them in the improper size ...or even check for them at all.
We've always carried a cheap pair of cables in the correct size just in case. We don't intend to ever actually use them. I've installed chains on enough trucks and buses during my working life to know that it's not fun when it's freezing, muddy, and wet to be out there wrestling with installing chains or hassling with them when they get wrapped around between the duals. Plus, chains can do a lot of physical damage to a motorhome when cross-links break. Also, some motorhomes have clearance issues around the wheel-wells where chains just aren't practical to begin with.
No, it's just not worth hassling with chains on a motorhome ...and don't forget you also must adhere to the minimum chain requirements for the toad too when dragging it behind you as it's legally considered a "trailer" when it pertains to minimum chain requirements.
There are numerous "Snow Zones" in Oregon both on state and interstate highways. The most well-known ones are on I-5 in from Roseburg south to the California border (Siskyou Summit and other passes preceding it) and in the northeastern part of the state on I-84 between Pendleton and Ontario.
They'll be designated with a sign like this:
Again, by law, you're required to carry chains in the correct size whether they intend to be used or not. Should you decide to break the law and not carry them, as some have mentioned, the probability of getting checked is very remote but I would strongly recommend that you do carry a cheap pair even though you have no intention of using them because of the reasons previously mentioned.
Here's where you have to carry them in Washington state (skip down to item "i" for the list):
WAC 204-24-050: Use of tire chains or other traction devices
Again, Washington has similar laws to carry chains during the general time period of November through March when using the roadways listed in the link whether you intend to use them or not. Both Oregon and Washington can extend the dates by requiring chains before November 1st and later than March 31st depending on conditions.