There are a number of passes through the western states that require chains for travel during inclement weather. They do enforce when the weather is bad. When I travel down this time of year, or later - I carry a two sets of cable chains; one for trailer and one for truck (SRW). I have yet to ever put them on - I choose to stay over at the base of a pass, if the weather is bad - then go over when it gets sunny. I plan this flexibility when I travel.
The requirement varies from state to state, but in general, it is one set on the drive axles and one set on a dual axle trailer, when the trailer has brakes (true in all of our scenarios).
No trouble with campgrounds open as you get south. A little planning is required, but normally most interstate accessible CG's are open - you may not get water at every site, but you will get electricity and have to use dump station on the way out, if you choose.
Hope this helps.
Brian
PS - I am not on the Island, but have in-laws in Cobble Hill and Duncan - been there lots. This year we are travelling down through Osoyoos (highway 97) near the end of this month, travelling over to Utah and then into southern California through Vegas. I suggest getting over that way and going through Vegas, Needles, Havasu, and Yuma - to see that territory. You may find Yuma a little windy, depending where on the Island you are from. We choose El Centro (farming area) as it is a little less Metropolis - and the weather is a bit better in the Imperial Valley in the early season (January, February). Have fun!
PSS - Make sure you know the border rules regarding what you can bring across - we choose to bring no meats or vegetables across the US border and stop across the border and shop. Same for pets if you are bringing the furry ones, make sure you have the appropriate papers (ie rabies vaccination confirmation) before you cross.