Is it possible for electrons to quantum tunnel through charged aluminum foil?
Yes. Quantum tunneling shouldn't be thought of as movement "through" something, exactly, as the name might seem to imply.
Various dynamic characteristics of electrons (really, all matter) are governed by statistics. Quantum mechanics then uses mechanisms like wave possibility functions to model these statistics. The convenience of this is that these wave functions can interact constructively or destructively with other wave functions; in these interactions, probability (the square of possibility) is conserved much like how energy is conserved when acoustic or electrical waves interact.
That being said, in any system, an electron's position can be predicted with a statistical distribution of possible positions. The position of an electron does not exactly exist (using the words of one version of quantum reality) until being measured. When measured, we discover exactly where the electron exists. The next measurement may show an electron somewhere else, and that exact position will be unknown until measurement, but we know that it will fall somewhere on the probability curve that we've found analytically.
So now imagine an electron in a box or a hole. The depth of this box is infinite. That means that regardless of how much energy the electron has, it could never hope to be able to exit the box. If the electron was a ball, regardless of how much energy it had to roll up the side of the hill, it would never reach the top of the hill and would thus never find its way out of the valley.
Now make that box a little more realistic and simply make its walls pretty high instead of infinitely high. Here, it is possible that some electron may one day have enough energy to find itself outside of the box. However, there's no way to tell when or how it got there. We simply know there is some possibility (positive probability) that the electron will be outside of the box.
Over enough measurements, that means that eventually an electron will be found outside of the box. We say that the electron has "tunneled" out of the box, but it never really existed inside the box. It only had a probability that it could be inside the box, and this particular electron, when measured, was one of the few that were outside the box.
So in your example, the aluminum foil would need to be charged with INFINITE potential in order to keep electrons from tunneling through it. Otherwise, if an electron could possibly have enough energy one day to find itself outside the aluminum foil, then one day electrons WILL be found on the other side of the aluminum foil. As you increase the charge, fewer will be found there as often... but still some will be found there. The ones that are found there shouldn't be thought of having once existed on the other side of the foil and happened to move through the foil... They should be thought of as perhaps always existing outside of the foil or possibly existing on either side of the foil. They just collapse to one possibility when you measure outside the foil.
I recommend a book by Richard P. Feynman (QED: The Strange Theory of Light and Matter) who uses a different quantum mechanical model to explain the same phenomena. In his model, he uses a sort of "all possible paths" approach to modeling quantum probability. It produces the same answers, but perhaps motivates a different sort of intuition. It's a short collection of lectures. I recommend reading it.
Ted Pavlic, B.S., Electrical Engineer, The Ohio State University
'I beseech you to take interest in these sacred domains so expressively called laboratories. Ask that there be more and that they be adorned for these are the temples of the future, wealth and well-being. It is here that humanity will grow, strengthen and improve. Here, humanity will learn to read progress and individual harmony in the works of nature, while humanity's own works are all too often those of barbarism, fanaticism and destruction.'