Is there a more elementary example of the holographic principle?
Disclaimer: I'm not an expert on the holographic principle. I'm posting this answer because it might have some limited value, but I hope somebody else steps in to give you a real answer.
is it the case that knowing the electric and magnetic field on the surface of a sphere will tell me the spatial distribution and velocities of charges within the sphere?
No.
For a counterexample, consider three concentric shells: The innermost one has static electric charge $+Q$ spread uniformly over its surface, the middle one has charge $-Q$ spread uniformly over its surface, and the outer one is where you make your observations. According to classical electrodynamics, the electric and magnetic fields on the outer surface are both zero for any $Q$, so value of $Q$ is not encoded on the boundary.
The holographic principle is different, and despite the name, it's also different than an ordinary hologram. In an ordinary hologram of an opaque object, you don't see the inside of the object. In the thing called the holographic principle, the lower-dimensional encoding is all-seeing, and the possibility of such an all-seeing lower-dimensional encoding is closely associated with impossibility of cramming unlimited amounts of information into arbitrarily small spaces in the bulk. That limitation, in turn, is closely related to the fact that massive objects automatically bend spacetime, the phenomenon we know as gravity.
(Caveat: the question of whether or not a "holographic screen" can encode what's inside a black hole might still be unsettled, but the question of whether or not a black hole really has an "inside" that's informationally independent from its "outside" might also still be unsettled. ...or maybe they are settled and I just haven't learned how yet. I have a lot to learn.)
Is there a more elementary example of the holographic principle?
The the AdS/CFT correspondence is most well-developed family of examples we have that exhibit the holographic principle, but even the simplest examples of the AdS/CFT correspondence (like AdS$_3$/CFT$_2$) are still far from simple by my standards. If a more accessible example of the holographic principle does exist, I hope that somebody else posts an answer about it, because I'd love to learn about it.