Can you vibrate an object and split its molecules? Let’s say I have paper (or, I don't know, any other material). Can I with some sort of device vibrate the molecules of those atoms so fast that they separate, and as a result I don’t have paper anymore but just air?
If so what is this called?
Please try to simplify your answer.  
 A: Well, any mechanical vibration (like a paint shaker) would tear the paper before tearing its molecules, and you'd be left with little bits of paper that wouldn't interact with your shaker.
What you'd have to do to break a molecule apart into constituent atoms would be to somehow tug on different parts of the molecule in different directions.  One way to accomplish this differential vibration would be to set up an electric field near the molecules, which pulls on the positive nuclei and the negative electrons in different directions, and then change the direction of that electric field very quickly.  An electric field that changes direction very quickly sets up an electromagnetic wave.  It turns out that most atoms, and most molecules, resonate at special frequencies and will respond very dramatically if you vibrate them at just the right rate.  The way your microwave oven works is to set up an oscillating electric field at just the right energy to vibrate water molecules.  However, that doesn't destroy the water molecules; what happens is that their vibrational energy spreads to other molecules nearby, which your body interprets as heat.
I'm assuming you have in mind a coherent vibration, rather than the random jumbled-up vibration that is heat.  Otherwise, as a commenter says, you could just hold your paper up the the vibrating molecules in the air around a lit match.
The problem with using electromagnetic fields to vibrate molecules is that each molecule can extract energy from light only in lumps.  For any reasonable amount of electromagnetic vibration, a particular molecule is much more likely to transfer the energy it picks up back to its environment than to stay excited and absorb another lump of energy.  (The technical name for a lump of energy that comes from an electromagnetic field is a photon.)  So to break apart the molecules in paper using an electric field, the field would have to oscillate rapidly enough that a single lump, or photon, could knock an electron away.  Those frequencies of light have the color that we call "ultraviolet."  Ultraviolet light (and its higher-energy cousins, x-rays and gamma rays) are dangerous biologically precisely because they can dissociate complicated molecules into their constituent molecules.
You have probably seen a piece of paper whose color has changed due to long exposure to ultraviolet light; if not, a docent at your favorite art museum might be able to show you one.
So the answer to your question is "sort of, but quantum mechanics gets in the way."
A: What you're describing is the end result of increasing thermalization, or the addition of energy to the mechanical degrees of freedom of a molecule. These degrees of freedom include rotational or translational ones as well. Energy stored in these degrees of freedom is called heat.
So you're not describing a process that is at all exotic. As Chris, one commenter, put it:

A match would suffice :) 

The breaking of bonds as the end result of energy addition is very common in chemical reactions. The endings "-lysis" "-lyzation" from Greek "Luein" to unbind (c.f. German lösen (unravel, solve, dissolve), English "loosen" and many other modern Indo-european derivatives) in names for chemical and biological processes often (but not always) betoken exactly this kind of thing.
