In the books we learn that James, Sirius, Pettigrew all became animagi while at school. Professor McGonnagell was obviously an animagus from well before Harry Potter was even born.
Did any students at Hogwarts become animagi during the events of the Harry Potter novels?
Nope. Of the named animagi, all of them were capable of transforming before Harry Potter was born and of the remaining (unnamed Animagi), none appear to have registered during Harry's school years.
We've since learned (via pottermore) that a team of 14 year olds from the Uagadou School of Magic have
"recently"
demonstrated their ability to turn into animals such as elephants and cheetahs but obviously these aren't Hogwarts students and hence don't count toward your question.
