Can I register and operate a nonprofit business though I move around a lot?

You can choose where you want to form your non-profit so long as you have a street address and a registered agent in the state of formation. In forming a not-for-profit corporation, you will want to comply with both state and federal laws. Compliance with federal rules and regulations will insure that you don't wind up owing more tax on your proceeds than you would otherwise. Compliance with state rules and regulations will insure that you have correctly formed your organization and that you are authorized to carry out business in that state.

If you will be carrying out the business of your non-profit in states different from the state of formation, you may need to comply with local and state business licensing requirements. In legal terms, you may need to qualify your non-profit to do business in more than one state. In many instances, a local attorney or one specializing in forming for profit and not-for-profit corporations should be able to help you. Good luck!

Here's just a couple of thoughts about your post: you'll want to establish a principal place of business within Washington that isn't dependent upon where you are living. You'll also want to seek someone or pay one of the services to act as registered agent in Washington. Sounds interesting...good luck!