The right to require Covid-19 vaccination in places of business, entertainment, and education ultimately depends on state or local law. So be sure to check requirements specific to your area.
But generally, the answer is yes. Businesses that provide in-person services may require that you be vaccinated before being able to enter their store or office. Places like sports arenas and concert halls may require the same of their customers. As an employee, your company may also ask that you get the shot before you return to work.
Similarly, summer camps and private schools may require children over the age of 12 to get the Covid-19 vaccine before arriving on site. That may be the case in public schools as well, which would fall in line with longstanding vaccination standards. In most states, children already need to get certain shots before being allowed to go to school.
Of course, there are always exceptions available for those who need them. People with certain medical conditions, for example, may not have to follow vaccination requirements.