The question is, how do you calculate the normal vector that the paddle will have to reflect the ball off of, in order to make the ball go from traveling in directionA to traveling in directionB

If the ball is traveling in direction A, it has the velocity vector A (dx, dy). Then, the desired resulting direction is the velocity vector B (dx, dy).

You can then get theta (angular direction) with tan(dy/dx).

The paddle's normal should be the average of (-theta of A) and (theta of B ).

Normal Angle =...
= (-(theta of A) + (theta of B )) / 2
= (-tan(A.dy/A.dx)+tan(B.dy/B.dx))/2

I think

Make sure to check for cases where dx == 0, since this will result in a divide by zero!

This should work...

I' = I - 2 * dot(N, I) * N

Where I = incoming direction vector (normalised) and N = normal of the paddle (normalised)