The general answer to this is: no, no license is needed - at present. You must however comply with all local regulations on business operation and taxes. A local accountant is the best person to advise on this.
In the US, it is worth bearing in mind that the FDA can legally consider your sales as tobacco products now. Although this is not exactly cut and dried, since there are no specific regulations in place, theoretically if you have some sort of problems then this might also come into the equation - perhaps brought in by a zealous city official trying to rack up offences.
In the UK, you need no license whatsoever to sell any products of this type although you will be subject to materials testing and analysis at the local level by your city / local Trading Standards inspectors. As yet e-cigs etc are not classified in any way except as general sales products.
This applies even if you only sell online, if you have an office in the UK. The officials in the area your office is located in will visit and take away whatever they want to, and inspect / analyse it. There are regulations that apply to electrical products such as chargers; to packaging of toxic materials (currently called CHIP regs but just about to change to a new system); but nothing at all for
e-liquid except general food regs, i.e. no known poisonous contaminants are allowed. They will analyse the liquid. An e-cigarette by itself has no regulations of any sort applicable to it - kind of like a flashlight.