First of all, I think this is a weird question but I still want to know whether it's possible. There is a character in a cartoon network series who is really really fast. Sometimes he lets his own body vibrate really hard so he can go through for example a wall. My question is: Is it possible for materials to go through other materials when they are vibrated fast enough?
I'm afraid that outside the cartoon world vibrating won't allow a solid object to pass through another solid object.
Solids won't pass through each other because the electrons in one solid interact with electrons in the other solid. There's a question about this already, How can I stand on the ground? EM or/and Pauli?, and there are lots of answers to it discussing the issues involved. Basically electrons don't like to be squeezed too closely together, and to squeeze two solids into occupying the same space would take a ridiculously large amount of energy. That much energy would simply blow both solids apart.
There is a sense in which you can pass matter through solids. If you take a very thin gold film and fire subatomic particles at it most of the subatomic particles will go straight through. This experiment was done in 1909 as part of the study of atomic structure. However while individual particles can pass through without hitting anything, anything bigger, even something as small as an atom, is very unlikely to make it through unscathed.
If you are talking about rigid macroscopic materials which is no doubt the type depicted in the cartoon, then no. There is nothing in the vibration of someone's body, for example, that would change any property of a rigid wall; you can't escape needing to break bonds between atoms in the wall (i.e. destroy it) if you want to pass through it, simple by doing something to your own body.
Of course, there is a caveat to this in the sense that if you were to vibrate a certain hard material (like steel for example) fast enough, then you could, in fact, break through a wall, which is going "through" it in some sense; but I doubt this is what you mean.