Grandmaster Yoda himself was capable of summoning force lightning in the form of Emerald Lightning. Shaak Ti utilized this Force power. Her killer, Galen Marek was an extremely skilled user of this Force power, even after his conversion to the light side.
It's a light side ability called tutaminis. It's the same thing that Vader uses to absorb or deflect blaster bolts with his hands. Like when he absorbed Han Solos bolt and disarmed him through the force. Although Yoda is basically the master of tutaminis, considering he was an avatar of the light side.
He tried, but was unprepared for the sheer power of Sidious' blast. It was magnitudes of order more powerful than Dooku's lightning, and it overwhelmed the defenses Yoda tried to throw up against it. Yoda just had no idea it would be so much stronger than what Dooku was able to muster.
Yoda was incredibly powerful with the force (one of the top 5 force users of all time for sure) and he was able to block the lightning and push in back to Palpatine, which surprised the latter greatly. Force Lightning is a sith speciality, so it won't make sense for Yoda to teach Luke how to conjure one.
There are two possible reasons for this; Kylo was never taught the skill or he simply chose not to use it. Rey, on the other hand, struggled with the temptation of the dark side at that moment and was fighting a connection to her grandfather, unbeknownst to her.
Darth Vader can not use force lighting, because of his severed arms. Robotic hands can't summon the force lighting. However Darth Vader can still use other forms of the force, such as force choke.
Some members of the New Jedi Order such as Luke Skywalker (who also used the dark side version at times), Jacen Solo and Jaden Korr also used light side Force lightning known as "Emerald Lightning", or Electric Judgment, but it remained somewhat controversial to some Jedi.
Although, with the battle against Mace Windu in his office, Palpatine's Force-lightning was reflected back and scarred his face, making it wrinkled and appear far older than he already was.
Why didn't Palpatine just stop his Force lightning when Mace Windu deflected it back at him? It was all part of his manipulation of Anakin and the Senate. He needed Anakin to think Mace Windu was executing a helpless prisoner, and he needed the Senate to see evidence that the Jedi were traitors.
It seemed Yoda had it all as a Jedi, but he also had a major weakness that most Jedi Masters shared: complacency. Yoda and his peers were too used to the Jedi Order's success. Yoda's complacency meant he was too late to take on the Dark Side until it was too late, and Darth Sidious had already risen to power.
How powerful would Yoda be if he turned to the dark side?
Along with force lightning and force choke, there is also the brutal force rage. If Yoda had been able to use these powers in combat, he would've been extremely difficult to beat. With the connection that Yoda had to the force, the impact of him using dark side powers would mean unparalleled destruction.
Dooku first used Force lightning at a young age. According to the audiobook, Dooku: Jedi Lost, he was once abducted for a Dark Side ritual, conjuring lightning to kill his captors and escape. Dooku went on to become one of the greatest Jedi masters, but his ability to use lightning was seemingly a natural talent.
When it comes down to both combatants in their prime, Yoda was superior to Darth Vader. That debate extends even further to prime Anakin Skywalker as well. He would have been outmatched by Yoda for many of the same reasons Obi-Wan outmatched him, and more.
He was fully in control of the fight when he was blasting Luke. Instead of trying to outright kill his enemy, Palpatine just used enough force lightning to torture him for his entertainment. Not only that, but Luke didn't have to endure it for as long as the Sith Lord.
Some have even speculated Sith can grow addicted to Force lightning. This naturally explains why Jedi do not generate Force lightning. Jedi seek to master their emotions rather than be mastered by them; Force lightning is therefore forbidden, as it is best understood as a spontaneous outpouring of hatred and rage.
Anakin Skywalker's Chosen One Potential Was Permanently Stunted. The Darth Vader comic book series reveals that, while the eponymous Sith Lord was indeed capable of using Force Lightning, the electricity could potentially destroy his life-sustaining suit.
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.
Blue in color, Sith shoot Force lightning from their hands by calling on their hatred and aggressive feelings. However, while a deadly weapon, it is not unstoppable.
Due to Vader's mechanical suit, he cannot interact with lightning under the pain of suffering malfunctions in the vital life support systems he uses. This is a rare problem for him to come up against, due to only other Sith commonly utilizing such a capability.
