The true Force-form of Darth Vader explains why he can't use Force Lightning. Conceived by the will of the Force itself, and with a midi-chlorian count so high it was greater than even Master Yoda's, Anakin Skywalker was the greatest Jedi Knight of his generation. His heroism was proved time and again as he fought on the front lines of the Clone Wars, tearing through Separatist forces relentlessly.
As great as Anakin Skywalker may have been, he was sadly vulnerable to the dark side; he became the legendary Darth Vader, a powerful Sith Lord. His body ravaged from a fateful battle with Obi-Wan Kenobi on the planet Mustafar, Darth Vader became more machine than man, and yet according to Emperor Palpatine he was still "unparalleled." Darth Vader proved his might by hunting down Jedi survivors of Order 66, slaughtering them with ease. Although he was regularly targeted by forces such as the Rebel Alliance and Crimson Dawn, he was rarely defeated - and he always fought back. And yet, for all that's the case, there were powers Darth Vader was unable to use - including his master's favorite trick, Force Lightning.
Darth Vader #8 - by Charles Soule, Giuseppe Camuncoli, Daniele Orlandini, David Curiel, and Joe Caramagna - hints why Vader cannot use this power. It reveals how Darth Vader appears in the Force - as a swirling maelstrom of the dark side, surrounded by tiny 'butterfly' glimmers of the light side he will later re-embrace. Strikingly, though, his cyborg limbs are presented as unconnected to the Force. Jedi and Sith can access the Force in part due to the influence of midi-chlorians in the blood, so it makes sense that Vader's cybernetics can't channel the Force (though this may also be because Vader's bitterness and self-loathing prevent him from seeing Palpatine's additions as true parts of himself.) 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. The mysterious visions in Darth Vader #8 confirm this, and further hint there are probably other Force powers Darth Vader cannot use for the same reasons. Indeed, even Darth Vader's greatest powers - such as his Force telekinesis - appear to have limits, ones he likely would not have possessed if not for the grievous injuries he sustained at Obi-Wan's hands.
Given Darth Vader's obsession with power, the fact that the harm inflicted by Obi-Wan limits him in this way - or at least that he believes it does - is likely another ingredient in the roiling sea of resentment he feels towards the Jedi. Hopefully, Marvel's comics will continue to explore this aspect of Darth Vader's limitations, as it's a fascinating opportunity to explore the Force's relationship to midi-chlorians and advanced technology.