What your state adds
Updated
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The federal determination decides your label. Your state decides whether you may operate at all, and the two are independent.
Federal is not the whole picture
Exempt soap sits outside FDA's cosmetic authority and under CPSC.
Neither of those is a license to manufacture, and neither says anything about your kitchen, your business registration or your sales tax.
Those are state and local, and they vary enough that a maker two states away has genuinely different obligations.
What states actually regulate
Business registration and the entity, which is universal.
Sales tax registration and collection, including at markets, where the organizer frequently requires proof of it.
Cosmetic manufacturing registration, which some states require and most do not.
Home occupation rules and zoning, which is the one that reaches a maker working from a residence.
And market or fair permits, which are set by the organizer and the local health authority.
Where soap gets caught
Not in the product rules, usually, but in the premises rules.
Cottage food laws generally cover food and do not reach soap, so a maker who assumes their state's cottage food exemption covers them is usually assuming wrong.
Handling sodium hydroxide at volume in a residence can also reach local fire and building requirements, which are the rules nobody thinks to look for.
Finding out
Ask the state agency that registers businesses, and the local planning or zoning office, in writing.
Ask the market organizer what they require, since they know their own jurisdiction.
And keep the answers, because market organizers and shop buyers ask for them repeatedly.
The FDA small business fact sheet is a reasonable starting point for the federal side of the same question.