It is often said that "everything is in a state of constant vibration". What led to this statement? And can I get any source of this statement that I can cite?
Billy Jean's answer pretty much covers the most likely answer, but I'll answer the less likely one. Everything above Absolute 0, -273.15 degrees Celcius, 0 degrees Kelvin, and -459.67 degrees Fahrenheit, has molecules that are moving around. In fact, temperature is nothing but the average kinetic energy of the molecules. The molecules bounce around faster with hotter objects. This is how conduction heating works: as faster moving molecules bump into slower moving molecules, they make them have higher kinetic energy, heating up the object. Because of temperature, you can say everything above Absolute Zero is vibrating, though "vibrating" is usually used on a much larger scale(usually visible with the naked eye).
The following would be a very basic look at things being described by string theory. A theory of vibrations.
If you are to zoom with a microscope smaller and smaller, you eventually hit a size that is "too small". This is known as Plank Length, approximately~ 1.616199(97)×10−35 meters. After this size, things smaller, dont really make sense. At this size, it says that there must be strings, that make up the "fabric" of space/time (consider it one entity [ie. x,y,z,w]). It is these strings that are infinitely vibrating, oscillating energy throughout our unobservable universe. These strings, in theory, when vibrating infinitely, create "pockets" during periods of excited energy or from colliding strings. We, our current perception of time, is theorized to be one of these pockets.
Like a rock colliding in water, it ripples. Like that ripple, we are said to be experiencing the ripple of time. And like anything that ripples, it echoes. We are the "echoes" of vibrating strings.
Scientists have in essence been measuring these "ripples" aka vibrations, inversely, from the time of the ancient Sumerians. We have begun from the macroscopic measurement of the universe, arguably mapped by ancient astrology. To Galileo and Newton, measuring the ripples of space, by mapping the solar system and figuring out our rules governing the laws of gravity.
With current particle accelerators and the rate of exponential technological growth and computational processing, we don't have very much longer (in my opinion), before we have a strong grasp and understanding of our quantum critical world. Once we reach this point, I believe we will begin experimenting with larger "quantum" experiments that will give us a better understanding of our origins.
There are hundreds of scholarly articles correlating vibrations to the function of pretty much everything. We can describe almost anything as wave if we break it down. Here is a generic set of results from google scholar correlating wave functions to (X). Feel free to tinker with the keywords and see what things you would like to find correlating to vibrations.