He is only the most potent Force user ever.
He is a dark side practitioner.
The Emperor and others use it.
Why didn't Darth Vader ever use Force Lightning?
In the Legends continuity, he doesn't because he can't.
According to Rise of Darth Vader:
his arms and legs were mechanical, and he could never cast Force Lightning
According to the Revenge of the Sith Visual Dictionary, under Darth Vader:
As a result of having artificial arms, Darth Vader will never be able to conjure Sith lightning-nor be invulnerable to it.
The Book of Sith states that
The second school, or focus, Body, encompasses those abilities that draw from the living Force. They emanate from your own cells and affect the physical structures of others. For this reason, any cyborg limb or enhancement will hinder your ability to conjur the effects of Body.
It goes on to name Lightning as an example of a "Body" ability.
But it was. His promising new apprentice, who was to be the greatest Sith who’d ever lived — maimed and burned, perhaps dead. Darth Sidious ground his teeth in frustrated anger. Part of him wanted to turn on his heel and leave what was left of Darth Vader to burn to ashes in the rising lava. Even if he was alive, even if he could be saved, Vader would be crippled.
And not just with his mechanical limbs. The Force — dark side as well as light — was generated by living beings, and it took living flesh to manipulate it. Darth Vader would never be able to cast blue Force lightning; that required living hands, not metal ones. And with so much of his body replaced by machinery, he would never come close to the potential he’d had.
It was a great pity, Darth Sidious thought, controlling his anger, but perhaps not irreparable. Even diminished, Darth Vader would still be very strong, and there were no Jedi left to challenge him. Darth Sidious had seen to that himself. So he kept walking until he could bend over the body. And to his surprise, his apprentice was still alive.
This is backed up by the new (and fully canon) factbook "Star Wars in 100 Scenes"
Vader is skilled with a lightsaber, though his mechanical arms prevent him from creating Force lightning
Somebody mentioned short circuiting the armor suit.
We can assume he could have used force lightning, evidence shows he would have died in the process due to design flaws. This website mentions a variant Vader used in a video game.
Many people are saying that he couldn't because of his suit, but this is never confirmed in canon. It's like saying that Sidious couldn't use Force choke because he never did it in the movies (he did it in the clone wars).
There is no definitive canon answer yet in the movies, the Clone Wars, rebels (tv show), nor the Marvel comics (Vader, Vader down, Star Wars...) or new books. These are the only canon stories until now. They might explain it later though. Just because we haven't seen it, doesn't mean he can't.
Before the Clone Wars Sith magic was never mentioned (in canon) either, and yet Dooku and Sidious used it. The reasoning that X can't use power Y because we haven't seen it is wrong.