Can randomness exist? Considering every cause has an action, how can anything be random? For something to happen, it must have a cause and through that definition it can't be random.
Considering this why are many quantum mechanical phenomena attributed to "randomness"?
 A: You should look at the link that Qmechanic gives, as it is closely related to your question.
The "randomness" in quantum mechanics is widely misunderstood. There is nothing random in the wavefunction (or quantum field theory description) and as long as all interacting systems stay entangled the behaviour is completely predictable. We only see randomness when the system decoheres, which typically happens when when we make an observation.
The randomness when making an observation is normally considered built into quantum mechanics i.e. there is no explanation for it. It's just the way quantum mechanics works. It may seem a bit unsatifactory to just have to accept the randomness, but remember that quantum mechanics is only a mathematical model - one that so far successfully describes the universe we see around us. All mathematical models are based on some assumptions, and the randomness in making measurements is one of these assumptions. It's possible that some deeper model will be developed one day, and this model will explain why the randomness occurs. However there is no such model at the moment, nor even a hint of one.
A: 
"Considering every cause has an action, how can anything be random?"

That's not really true. What happens is lots and lots of things happen at random and 'on average' you get the same predictable result.
Even classical areas like thermodynamics and fluid dynamics are really just the statistical
behaviour of lots of random individual effects.
A: Randomness is just a word to compensate our lack of proper understanding. It's logical that actions cause reaction. Everywhere and always.
