If you mean, can the Washington LLC operate in California? It would very likely need to register there first.
This is AVVO, a place for users to obtain general legal information to general legal questions. I am glad to help you in any way I can, within those limits. I wish to make clear I am only communicating with you for the sole purpose of exchanging such general information, and nothing more. It is not legal advice, which I can not provide because among other reasons I know few of the necessary details of your situation. I do not purport to represent you in any way, shape or form. Of course, if you would like to seek out my services, and if you are a NY resident, I will probably not put up very much resistance but representation would still necessitate a signed retainer agreement between yourself and I. Thank you.
Mr. Stumpf is correct. If you're "doing business" in a state other than your "home" state, i.e. the state where you formed the business, you will almost always have to register with that state's Secretary of State. "Doing business" usually requires some sort of on-going business within the state, e.g. opening a store there.
If you're question is getting at whether you will need to create an entirely separate entity to do business in another state, the answer is no. Your WA LLC can conduct business in any state, but it will be subject to each state's laws that govern LLCs. Therefore, you should review the LLC laws in each state where you're hoping to conduct business to determine what it takes for a foreign LLC to do business in that state.
If you're curious about any of the laws, or you're having trouble understanding what it will take to register your LLC in another state, you should contact a business attorney. Feel free to contact me directly at firstname.lastname@example.org if you have specific questions.