However, we were taught in my class that the dipole moment of a molecule is equal to the vector sum of the dipole moment of each bond in the molecule. But Ozone only has bonds between two Oxygens, so why is the dipole moment of each bond not equal to zero?
Chemical education is in such a sorry state. Why do they teach chemistry like chicken or eggs story? Did the chicken come first or the egg? I have yet to find a single general chemistry book which even mention how the dipole moment is measured (even in one line). If you want to pursue science, instead of thinking about imaginary fairy tales, ask the teacher how would you experimentally measure the dipole moment of a gas? How do we know that a given gas has a permanent dipole moment?
The dipole moment of gases is mostly determined by microwave spectroscopy. In chemistry, experiment usually comes first (unlike chicken and egg). So if you shine microwaves on ozone, ozone would show a rotational spectrum. This is a signature that yes ozone must have a dipole moment. Now you have to consider a consistent electron distribution which would lead to a permanent dipole moment.
Similarly, electron diffraction experiment will tell you that ozone molecule is not linear. When it is not linear and it is rather bent, and yet it has a dipole moment, now people have to think of the electronic distribution.