Rules differ by state and sometimes by city, so this guide covers the questions to ask rather than a single answer. It is general information, not legal or tax advice. Your state's department of revenue, or a local accountant, has the final word for your situation.
Most states that charge sales tax require anyone selling goods to register first. The document is called different things depending on the state: a seller's permit, sales tax license, vendor's license, certificate of authority or transaction privilege license. Registration is normally done with the state department of revenue, often online.
Many organizers ask for a copy of your permit with the application, and some states issue a short-term or temporary permit for vendors who only sell at a few events a year. Ask your state which option fits.
Delaware, Montana, New Hampshire and Oregon do not charge a statewide sales tax. Alaska has none at the state level, but some Alaska cities and boroughs charge their own. Even in those places you may need a local business license, and you may owe tax in other states if you sell there.
Start with your state department of revenue and the show organizer. A local Small Business Development Center or an accountant who works with small sellers can walk you through the first filing. Mention that you sell at craft shows, since many have a standard answer for it.
Save listings, keep application deadlines in one place, and read what vendors who have worked a show say about it before you commit to a booth fee in your area.