Although hairstylists have an advantage selling hair because of a built in clientele, no, you don't have to be a licensed stylist to sell hair extensions.
Selling anything online will require you to pay sales tax, file taxes, and other actions required by all businesses, not just a hair business.
If you have experience setting up a business you should be able to get up and running rather quickly. We don't provide advice related to accounting or legal, that is what accountants and lawyers are for.
Most clients will set up an LLC, get a Tax ID, business bank account, and be off selling in no time with limited startup costs.