Cockatiel, are you asking if it's legal for a minor to do business? For examples of students who have done this, look at the threads in the financial aid section by students who applied for the McKelvey Foundation scholarship for student-entrepreneurs.
I think that depending on the kind of work, a sole proprietor may not have to take special steps to start the business. When I've done consulting as a sole proprietor, I've just worked and then reported my income on Schedule C of my tax return.
What is different about your son's situation is that he has a retail business, so there are issues of state sales tax, possibly county and city taxes, and county and city business licenses. If your son's own name is part of his business name, then he may not have to file a fictitious business name statement. That's how it is in my county. But he will need to check with the state, county, and city to see what they require. At the federal level, he might or might not have to get an EIN. As a sole proprietor you usually don't need one (you can use your SSN), but the retail aspect might change things.
My state has a nice collection of web pages about how to start a small business. You could check to see if your state has something similar.