Why is impulse a vector?

MOmentum is a vector and impulse is the change of momentum.
Impulse is the change of momentum. It is possible for momentum to change such that the momentum of an object increases, decreases or reverses direction. As impulse measures those possible changes it must be able to account for the possible direction by being a vector.
During this elastic collision the momentum of small mass changes to the left. But the momentum of the large mass changes to the right. So the impulse of the small mass is to the left and the impulse of the large mass is to the right. One must be negative and the other positive.
additionally impulse must be a vector to satisfy Newton's Third Law of motion. Consider the equation relating impulse, force and time $\Delta p=F.\Delta t$
If body $A$ exerts a force on body $B$ then body $B$ must exerts an equal and opposite force on body $A$. If the forces are equal and opposite then so must be impulses. That is not possible if impulse is a scalar