Orthogonality between $\vec{E}$ and $\vec{H}$ waves with space-dependent amplitudes I am able to prove in a few lines that the electrodynamic field vectors $\vec{E}$, $\vec{H}$ and $\vec{S}$ are all orthogonal to each other considering that $\vec{E}$ and $\vec{H}$ are coherent plane waves with constant complex amplitudes propagating in the $\vec{k}$ direction. See for example JACKSON J.D. Classical Electrodynamics (§7.1, Plane Waves in a Nonconducting Medium, 1st ed. (1962), pp. 204-205, eqs. 7.9-7.15).
My question is: is there any way to prove the same but considering waves of amplitudes varying with space $(x,y,z)$?
I haven't seen this even mentioned in any of the books I've consulted, they all consider plane waves with constant amplitudes. I tried to prove it on my own using the different methods I've seen for constant amplitudes but they all fail.
 A: As Ondřej points out, $\mathbf{S}$ is orthogonal to both $\mathbf{E}$ and $\mathbf{B}$ as it's defined as their cross product. However, $\mathbf{E}$ and $\mathbf{B}$ are in general not orthogonal to each other.
This is obvious in the general case: you can make static fields using Helmholtz coils and electrodes, completely independent of each other, for which $\mathbf{E}\cdot\mathbf{B}$ is completely arbitrary. In a sense this is trivially true and not interesting to the (radiation) regime you're asking about. However, it does make it clear that you need to be very precise with what sort of fields you are allowing, and what general premises you're starting with, because if they are too broad then the result will not be true.
Also, I'd like to point out that there's no such thing as "plane waves of amplitudes varying with space". Plane waves are plane waves, period. The most general plane wave possible has electric field
$$\mathbf{E}(\mathbf{r},t)=\rm{Re}\left(\mathbf{E}_0e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}\right)$$
and it is specified uniquely by a real wavevector $\mathbf{k}$  and a (possibly complex) polarization vector $\mathbf{E}_0$ such that $\mathbf{E}_0\cdot\mathbf{k}=0$. The amplitude stays constant and the wave spans all of space.
What I think you're after, though, is monochromatic fields, which are much more general, and which can be written in the form
$$\mathbf{E}(\mathbf{r},t)=\rm{Re}\left(\mathbf{E}_0(\mathbf{r})e^{-i\omega t}\right),$$
with a similar equation for $\mathbf{B}$, and for which the Maxwell equations are the standard ones if you replace $\mathbf{E}$ and $\mathbf{B}$ with the possibly complex $\mathbf{E}_0$ and $\mathbf{B}_0$, and all temporal partial derivatives with $-i\omega$ (and where you need to impose, because of the above counterexample, the nontriviality condition $\omega\neq0$). If this is indeed the régime you're after, then I'll try to think of a proof or a counter-example.
