An online business is no different from any other business when it comes to licenses and permits. Depending on the type of business you conduct, online or offline, you may need certain permits and licenses from three different authorities: the Federal Government, your State Government or your local authorities.
Federal Permits and Licenses
Federal Government permits and licenses are rarely necessary, except for certain types of rather dangerous activities, like drug manufacturing, selling of alcohol and firearms and so forth.

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State and Local Permits and Licenses
On the state level, as a New York business, you can find out what kind of licenses and permits are required by going to New York Business Express.
On the local level, as a New York City business, you can find information about permits and licenses here. As far as I can see, they do not require any particular license for the sale of new goods on the internet.
With respect to using your home as a home office in New York City, I wouldn’t worry about any permit or license, as long as your business doesn’t disturb your neighbors. But if in doubt, you should consult the New York City Zoning Code, or your town’s zoning code, if you don’t live in NYC.
By the way, don’t hesitate to call all State and City agencies and ask your way through to somebody who seems knowledgeable and get all the information you need. While there is no guarantee that people give you correct information (is there ever?), these government employees are being paid to help you (for free) and are often very helpful.
Then there is other paperwork that a new business may need to file, even though I wouldn’t call that a permit or a license.
Federal Identification Number (EIN)
For example, corporations and multi-member LLCs must apply for a federal identification number.
Sales Tax Authorization
If you are selling goods that are subject to sales tax, you have to get a sales tax Certificate of Authority.
Business Certificate
In addition, any business that conducts its business under a name other than the name of the individual owner, must apply for a “fictitious name certificate” or “business certificate.” In other words, Daphne Danzig selling widgets as Daphne Danzig does not have to get a business certificate. But Daphne Danzig selling widgets under the name “Ace Widgets” needs to apply for a certificate. If Daphne Danzig is in New York City, she can get her certificate at the county clerk, 60 Centre Street, for more info see here. For an entire post on business certificates, see my earlier post on “doing business as”.
**This post is for informational purposes only and does not constitute legal advice**