Answer
No.
Work Step by Step
The system started at rest. After the firing event, the cannon and the cannonball have equal and opposite momentum vectors. By the parallelogram rule, adding these gives a momentum of zero, which shows that momentum is conserved.
If momentum were not a vector, and we simply added the magnitudes (without paying attention to direction), we would find that system momentum was not conserved.
This is discussed on page 97.