Most likely your school considers its name to be a service mark, legally identical to a trademark, and you would need permission to use it on your shirts. (A service mark or trademark need not have a symbol next to it to be considered a protectable mark. Using the name in a particular context is sufficient. And even nonprofits can own trademarks.)

Your safest course is to get a written note from school officials allowing you to use the name for the T-shirts.