You would not open a brick and mortar store without advertising and hope to grow, and there is no reason to believe that starting an internet business without advertising will not suffer the same fate. When you build a website and place it online, you and your web host are the only ones who know about it. Without proper marketing, that is how it will stay until you finally get frustrated and shut it down.
The internet is not the Field of Dreams and if you build it, they will not always come. If they do it may be accident if you have not let anyone else know your site exists. Many people fail to properly budget for advertising and once their site is up, they are caught in the middle. They have no money to advertise unless they make some sales, and they cannot make any sales unless they advertise. Most take that as a good reason to give up.
Having T-shirts imprinted with your business name can be done fairly cheap and some places will do them on your shirts for as little as $5 or $10 each. Provided you stick to the basics. Wearing them to public events gets your name out cheaply. Volunteer as an expert in your area of expertise. For example, if you are a carpenter...