Are caracals legal in Washington state?
The Washington State Department of Agriculture’s website states that it is illegal to possess wild animals. The caracal is a wild animal. They list some of the animals which are illegal to own and in that shortlist is the bobcat. This reinforces my belief that the caracal is illegal to own in Washington state. The website also lists “dangerous animal law”. Under this law some animals are illegal to own. Surprisingly, under RCWs, Title 16, Chapter 16.30, Section 16.30.010, “Potentially dangerous wild animal” means one of the following types of animal whether bred in the wild or in captivity, and any or all hybrids thereof: Family felidae, only lions, tigers, captive-bred cougars, Jaguars, cheetahs, leopards, snow leopards, and clouded leopards.

Caracal kittens. Picture in public domain in my view.
Surprisingly, the caracal is not listed under this section. So the position is unclear on the internet. As usual, the legal documents presented on the internet regarding the possession and ownership of exotic animal species is nearly always confusing. The presentation of the information is poor and nearly always unclear.
Washington state say that it is illegal to possess wild animals and therefore the default position that it is illegal to possess a caracal. Thereafter, I would suggest that you ring up Washington Department of Fish and Wildlife and discuss the detail with them. No doubt they will get to the point very quickly and tell you the sort of permit you need in order to own and possess a caracal in this state.
Clearly, there will be circumstances under which it is permissible to own a caracal because there are zoos in Washington state where there are caracals. These organisations have permits. Different regulations would apply to individuals owning a caracal as a pet. You’ll need to know what those regulations are. There might be a blanket ban on owning a caracal as a pet in this state. I don’t know because their website is unclear on that issue. Good luck!
Cross-reference
I would cross-reference federal animal laws and more local animals laws i.e. city regulations, to make sure that there isn’t a clash. It is complicated I think. You have to be careful.
Owing a caracal as a pet
I have said it before which is that both the caracal and the cheetah are two wild cat species which are amenable to becoming “pets”. This is not to say that I personally recommend possessing these wild cats as pets. I don’t agree with it. Not at all because it is poor for conservation purposes. Keeping a caracal as a pet reinforces the belief that a wild cat species can be a companion animal which undermines their position in the wild. It encourages young people to believe that these wonderful creatures are ours to possess and do as we please with them. This is incorrect. Leave them alone in the wild where they belong. But protect them in the wild. Give them space in which to live natural. They have the right. That’s the best way forward. It is time that conservation was taken seriously by which I mean businesses and politicians should get involved with it seriously rather than dabbling in it and pretending.