The vector potential $A$ is perpendicular to $B = \nabla \times A$, by definition, and hence, in a plane wave, it is either in the direction of $E$ or the direction of propagation. I suspect it is in the direction of propagation.
What is its direction?
Answer
While for vectors $\vec{B}$ and $\vec{C}$, the cross product $\vec{B}\times\vec{C}$ is indeed perpendicular to both of the vectors, it is simply not the case that the curl of a vector field is orthogonal to the vector field. Do not read too much into the cross product notation.
In particular, you can add any constant vector field to $\vec{A}$ without changing the fields. So we can make it be nonorthogonal by adding a constant of our choice. When someone tells you a vector potential points in a particular direction they are simply making a gauge choice, and a different choice of gauge can result in the vector potential pointing in a different direction.
This means your question simply isn't well defined. We can find the direction of the electric field by seeing the force per unit charge of stationary charges, and we can find the magnetic field by finding the force on moving charges that move in three linearly independent directions. But there is no classical experiment to find the direction the vector potential points, so it isn't a scientific question.
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