It is not illegal, but I think you need to obtain the license from certain government department.
For selling online, you are in the grey area if you don't have the license and not register as a company or enterprise.
However if you want to integrate a payment gateway like MOLPay and iPay88, there are R&R you need to follow which may require you to register a company or enterprise anyway.

You can start an online business right away anytime.
Depending on how you want to do it and how big you want to do it.
Make sure the products you sell are legal and not against the law and regulation.

More and more complicated terms will come in when you want to integrate payment gateway in your website, so take note of it.