some tracks require you to wear the basic off road gear, such as riding pants, boots, helmet, goggles, gloves and a long sleeve shirt. But I have never heard of a track that requires you to ride with a chest protector on.....

you dont have to wear one but i would if i were you it protects your back and also protects your chest from roost

State where you are from as the country I am in it is required for Juniors to wear a chest protector, but the seniors do not have to although many of them wear armadillo or equivalent under jersey chest protectors. They are a great idea and protect the rider not only when they have a bad stack and perhaps get run over, but they help to protect from Roost as well.
So just post where you are in the world so you can get the correct information from someone who is in the same place.