A "seller's permit" or "tax license" (two terms for the same thing) is given by the State Board of Equalization here in California. I would assume that the name of the state department is the same in any state. What it does is allow you to purchase stuff genuinely at wholesale prices (not that very many places that offer art supplies in small quantities by wholesale), and without paying sales tax (not all businesses will do this, because they have to keep track of it and file when they do their taxes at the end of the year, explaining why THEY didn't collect sales tax for stuff they sold). It is also the license number by which you file YOUR sales taxes at the end of the year.
Sales taxes are a rather complicated thing, but the short of it is, if you sell things to someone and the good physically change hands in a certain location, then you must pay sales taxes to that location (many cities and counties in Cali have their own sales taxes tacked onto the state ones - not sure if that happens elsewhere). If you are in a state (like Cali) and mail a piece of art to someone else in that same state (like Cali), then you must pay the state sales taxes. That's why most conventions will require a seller's permit before allowing you to sell at the convention. They can get in trouble if they let people sell who are not paying taxes, and you can't pay taxes without that number.
Other required permits vary from city to city. Most cities require a business license for anyone doing business in the city throughout the year. Other cities will also require them for people who do business a certain number of times, or even once. San Francisco, for example, requires a hefty business license fee and special permit (last I checked, to the tune of $600) for the right to sell wares on the street for a single weekend (i.e. street vendor). If such permits are required in the city where a convention or other event is being held, they will inform you of the required permits, and where to obtain them.
So, check with the state government pages of Washington about sellers permits. If you live out-of-state, be sure to find out if they actually require one for you to sell just the one time. Some states don't think it's worth the paperwork for a one-time seller. Also, don't be afraid to email the organizers for the convention. Hopefully, they will be willing to give you some advice.
If anyone here has dealt specifically with Washington state, hopefully they can give more precise info. I've never sold there.
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