What should I do if I want to start an e-commerce site under a name different from the business name provided for my proprietorship firm? Should I apply for a new GST number?

GST Registration is PAN based and State specific registration hence with respect to your query there is no need to apply for new GST registration in case you want to start an e-commerce site. A person can do multiple businesses under one GST registration. However, for starting new business, an amendment is required in GST registration application by making addition of that business activity.