I always hear people talking very loosely about the terms "vibration" and "energy" in relation to crystals.
Is there any scientific basis in "crystal energy" or is it purely metaphysical pseudoscience?
Photons can travel through a crystal, so it can be a medium for energy to travel through, but that isn't exactly energy storage.
What type of energy can be contained within a crystal if any?
Crystals have internal energy including some energy associated with molecular vibrations.
Amorphous materials have internal energy including some energy associated with molecular vibrations.
In addition to molecular vibrations, crystals can exhibit mechanical (macroscopic) vibrations. An example of a resonant macroscopic vibrator is a tuning fork. But, note that most tuning forks are made of amorphous metals: crystals offer no special benefit here either.
A few crystals exhibit the piezoelectric effect: a mechanical stress induces an electric field and vice versa. This is not special to crystals. Polymers such as polyvinylidene fluoride also exhibit this same effect.
In sum, there is no special energy that crystals have that other materials do not have.
Googling your the keywords in your question, I found this site which claims:
Crystal energy is the power that crystals naturally give off and that you are able to use to heal yourself.
That claim is nonsense.
A second site says that the "energy" of agate, a merely cryptocrystalline form of silica, has the following properties:
Compels the truth. Promotes good manners, happiness, intelligence, prosperity, longevity, fertility and good health.
That is also nonsense.
As far as I know, there is zero physics in "Crystal Energy", which if I understand what you mean, is related to Crystal Healing. There is a good bit of validity to the placebo effect and belief, so a person believing in something, whether the practitioner or the patient, can have an effect or the impression of an effect, but that's not physics either. I've known a few people who believed in this type of crystal energy and I never heard a word of science come from any of their mouths.
There are, however, real types of crystal energy, such as, you might have heard the term Quartz Watch or Quartz time piece. This is because Quarts is a Piezoelectricitric material that has certain electric properties that respond to pressure and also, can keep very reliable time.
I've heard it theorized that quartz in the ground under tectonic stress can cause charges like lightning to come from the ground and what is sometimes called ball lightning - which is rarely seen and perhaps only recently, confirmed by science. Article on same, though, this is all pretty far removed from your question and I've never known anyone who squeezed a quartz crystal hard enough to get an electric charge out of it.
See, a crystal due to its temperature and through various media can be made to vibrate. Now because of the fact that the atoms(lattice points, could be molecules too) are connected with each other, the vibration actually spreads in all directions. Thus this vibration acts as an wave. But the whole of crystal, because of its structure can vibrate only in specific frequencies. So we thus have quantised waves within crystals, which can be treated as particles(due to wave-particle duality) called phonons and these waves carry energy.
Other thing is that when the atoms or molecules that made this crystal were very far apart they did not interacted. So once you put them close the coulomb interactions try to oppose the formation of crystal. So you need to do work to create a crystal. This workdone is stored within a crystal as potential energy.
