A standard fire arm should work fine.
Even though vibranium absorbs kinetic energy† and vibrations, the mechanism by which a standard fire arm operates is a build up of pressure behind the projectile. The projectile can be any substance you choose and vibranium will not detract from this effect to a degree that will significantly degrade the performance of the weapon.
Simply put, the Wakandan variety, simply known as vibranium:
[...] possesses the ability to absorb all vibrations [emphasis mine] in the vicinity as well as kinetic energy directed at it. The energy absorbed is stored within the bonds between the molecules that make up the substance. As a result, the more energy vibranium absorbs the tougher it becomes.
The significant difference between a shock or vibration and a pressure differential, in this case, is that vibranium will absorb the shock. Nothing is said about the effect of the pressure differential. Presumably this means that there is nothing interesting to note about it. In other words the effect of a pressure differential on vibranium is static, the same as it is on other naturally occuring elements - meaning a standard weapon should work as expected and the projectile will be ejected, even if it is made of vibranium.
However, one benefit (potentially also a caveat) of the bullet being made of vibranium is that the shot will likely be silent - without a silencing device:
There are limits to the capacity of the energy that can be stored, and although the exact limitations are not yet known, there have been a few examples. One such instance was when the oil conglomerate Roxxon discovered that a small island in the South Atlantic had a foundation composed of vibranium. Due to this, Roxxon found it necessary to destroy the island and so blew it up with bombs. Unable to absorb the force of the explosions, the vibranium was destroyed, but it did succeed in entirely absorbing the sound made by the explosion [emphasis mine], preventing damage to the surrounding area.
As long as the firing of the weapon is not powerful enough to destroy vibranium. It is not clear from the description how powerful the explosion was that destroyed the vibranium on the island.
† Presumably this is referring to impulse energy, rather than the more general term, kinetic energy.
As for special kinds of weapons, your guess is as good as mine.
But as I was reading about each kind of vibranium, the standard Wakandan variety as well as Antarctic variety, known as anti-metal, the idea came into the back of my mind about the possible interplay between the two isotopes.
The first absorbs and stores energy, making it tougher.
While the second actually produces vibrations which liquify other metals.
Presumably anti-metal does not liquify vibranium, but makes it stronger, however my suspicion is that there may be an interesting kind of oscillation or feedback between the two isotopes that could be exploited to produce very stange and powerfull effects.
The most obvious would seem to be a bomb, similar to a nuclear weapon, which by blasting the two metals together would produce an exotic reaction when the atoms of the two metals
fuse come into close, molecular proximity, one vibrating the other storing until it breaks, however the interesting part is what happens when the reaction begins.
Anti-Metal is able to become an artificial and unstable form of the Wakandan variety of vibranium through certain particle bombardments on it.
The result is explicitly unstable, however, we are talking about two exotic isotopes and the bombardment is coming from particles of an isotope of the same element. So it is not clear whether this would catastrophically explode or do something much stranger - like make a black hole perhaps? :D
One may never know...