What your locality allows
Updated
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A short answer, because the thing people go looking for does not exist.
There is no candle license
Candles are not food, cosmetics or drugs. No agency licenses candle makers, no inspection is required before selling, and there is no registration to complete.
What applies is the federal ban on lead-cored wicks, the consumer commodity labeling requirements, and a set of voluntary ASTM standards that nobody will check until something goes wrong.
What you actually need
A business registration and a sales tax permit, which are local.
A local business license where your city or county requires one.
Permission from a landlord, if you rent.
Insurance, which for this craft is the substantive one.
Fire code and quantities
Where the licensing question does bite is fire code, and it bites on quantities stored rather than on the activity.
Wax in bulk and fragrance oil in bulk are both combustible materials, and above certain quantities a commercial premises has storage requirements.
This is not usually a concern at a kitchen scale and becomes one at the point you take a unit and buy by the pallet, which is exactly the point at which people stop asking questions because they feel established.
The one that catches people
Home insurance. Melting wax over a heat source and burning test candles for hours is a business use of a domestic property, and an insurer who does not know about it may decline a claim.
It costs a phone call and it is the difference between a fire being an insured loss and a personal one.