Either way, while Vader can use most other Force abilities, Force Lightning needs to be channeled, and Vader's limbs are explicitly not connected to the Force. The Star Wars: Episode III - Revenge of the Sith Visual Dictionary confirms Darth Vader cannot use Force Lightning, and even suggests this is the reason why.
No, Darth Vader cannot use force lightning and never uses the ability in any of the Star Wars movies. Despite being a powerful Sith lord, Vader's cybernetic limbs and respiratory device are very sensitive to electrical discharge and therefore using force lightning was potentially life-threatening for the former Jedi.
And therein lies the answer. Sith specifically channel Force lightning using their hands. Seemingly, because Vader's hands are no longer there, he can't channel the dark side of the Force through them, meaning that he can't use Force lightning.
A lot of the systems that were keeping Darth Vader alive were maintained by electricity, which is why Force lightning was so destructive towards him. One of the main reasons why Darth Vader perished at the end of Return of the Jedi was due to the damage that was done to his system by the Emperor's Force lightning.
He hasn't learned how to use the ability yet, due to his still-shaky alignment in the Force. One must give themselves fully to the Dark in order to use lightning, and he just hasn't done that. Why did Kylo Ren not have a “Darth” name?
Grandmaster Yoda himself was capable of summoning force lightning in the form of Emerald Lightning. Galen Marek was an extremely skilled user of this Force power, even after his conversion to the light side, rivaling Darth Sidious.
1/15 Luke Skywalker
Having faced his father once before, Luke was familiar with his tactics and used the force to gain an upper-hand. In Star Wars Canon, only Luke Skywalker can be acknowledged as the one who truly defeated Darth Vader.
Because he was not an active member of the Jedi Order during Order 66, Infil'a escaped execution. Some time later, he faced the Sith Lord Darth Vader wielding a green lightsaber. Infil'a put up a fierce fight against Vader, almost killing him by throwing the Sith Lord off of a cliff.
His pride and arrogance consume him as they blind him from the realization that his actions aren't immune to consequence. His self-image of supremacy is instrumental to his crumbling; he cannot tear himself away from believing that there are other points of view besides his own.
By far, Skywalker's most infamous use of it was upon learning of the death of his wife, Padmé Amidala, after his transformation into Darth Vader. This caused him to involuntarily unleash a powerful Force scream of shame, rage, and grief, severely damaging the medical chamber and the droids within it.
In a Vanity Fair interview from 2005, George Lucas went deeper into why Anakin became much weaker as Vader. In the interview, he explained that because Anakin's limbs were severed and replaced by robotics, it was harder for him to use the Force than it was before his battle with Obi-Wan.
The Force: As the Chosen One, Vader was immensely powerful and had an amazingly strong connection to the Force, stronger than any Force-user ever in existence. He also possessed the potential to according to Obi-Wan Kenboi become the most powerful light and dark side Force-Sensitive being in the history of the galaxy.
The requirement of having an organic limb to conjure Sith Lightning is Legends only information. In Legends, Anakin never used because he didn't know how. In canon, it has been speculated that Anakin could, but never learned how to before becoming Darth Vader.
Vader held respect for Tarkin which is clearly shown and the only case in which Tarkin outranked Vader would be the case in which Vader was in the Death Star or within Tarkin's Oversector which he likely was as Tarkin was Grand Moff of the Outer Rim Oversector, the largest one.
This information comes to light form the comic series Darth Vader: Dark Lord of the Sith by Charles Soule and Giuseppe Camuncoli, in particular issue #7. Palpatine admits to Darth Vader that the Jedi he actually feared most was none other than Jocasta Nu.
Admiral Ozzel : [appearing onscreen with Captain Piett] Lord Vader, the fleet has moved out of lightspeed and we're preparing to... Darth Vader : You have failed me for the last time, Admiral! Captain Piett?
Did Darth Vader ever lost a fight? Darth Vader is one of the greatest villains of all time. But even he gets beaten sometimes. CBR lists all of the folks who managed this improbable feat!
Between Darth Vader and Obi-Wan, it's pretty clear that Vader is far more powerful with the force. After all, he pretty easily used the force to stop a starship from taking off in the previous episode, and manhandled Kenobi throughout the majority of the show.
The duel. Darth Vader defeats Darth Maul. The two Sith Lords engaged in a lightsaber duel, going through the volcano and onto a bridge situated over a river of lava, which collapsed, though they both survived and continued their battle.
Abrams' movie handles Force Lightning. When Rey taps into the dark side of the force on Pasaana and uses Force Lightning, she feels frustrated after apparently losing Chewbacca (Joonas Suotamo) to the First Order and being bested by Ben Solo/Kylo Ren (Adam Driver) in a Force tug of war.