I would write She is a third year student at (a or the) vocational school. She is doing a traineeship (and change traineeship to internship)at an advertising agency now.

You could write: She is a third-year student at (a or the) vocational school who is undergoing a a internship. Or you could write two sentences by splitting them in half: She is a third-year student at (a or the) vocational school. She is undergoing a internship.