We are given: $\vec a = 8\vec b, \qquad \vec c = -7\vec b$ where $\vec a, \vec b, \vec c$ are non-zero vectors.

Since $\vec a = 8\vec b$, the vector $\vec a$ is in the same direction as $\vec b$ because the scalar $8>0$.

Since $\vec c = -7\vec b$, the vector $\vec c$ is in the opposite direction to $\vec b$ because the scalar $-7<0$.

Therefore, $\vec a$ and $\vec c$ are in opposite directions.

The angle between two opposite vectors is: $\pi$

Hence, the correct option is: $\boxed{\text{D: } \pi}$