English generally omits articles for personal names, but some institutions may include it as part of their name, in which case it is always included but capitalization may depend on the style guide in use. In still other cases use of the article is conventional but not considered part of the name. So, the proper usage here basically depends on whether the institution is generally known as "Amerigo Vespucci School" or "The Amerigo Vespucci School," or if it provides guidance on its public identity.
She is a professor at The Ohio State University. She received her Ph.D. from the The George Washington University, and was prepared at The Lawrenceville School.
She is a professor at Kansas State University. She received her Ph.D. from George Mason University, and was prepared at Darrow School.
She is a professor at the University of Arizona. She received her Ph.D. from the College of Charleston, and was prepared at the Milton Hershey School.