In episode 50, Aichi's anguished rant at Kai. Effects of PSY Qualia aside, he was probably keeping most of what he said bottled up inside for pretty much the entire series...
Episode 102 has quite the Tear Jerker: Aichi arrives at Leon and Kai's fight as it finishes and rushes to Kai's side, whose initially unresponsive. Kai apologizes for having not been there to fight for the sealed clans by Aichi's side, while the whole time, he sits there in tears from just listening. When Kai, Ren and Asaka begin to glow from their dismissal from Cray, Aichi clings to Kai, pleading for him not to disappear and encouraged to fight on before they disappear, which is followed by Aichi giving a Skyward ScreamSay My Name and breaking down in tears completely before Leon. Even Charlene and Jillian feel for him.
Kai's feelings of inadequacy in Link Joker. He's had them for quite some time, at least since Episode 103, some months before the third season. They slowly begin to eat away at him and it becomes increasingly difficult for him to hide his real feelings, almost opening up to Miwa, though his words are mostly lost. Their worsening correlates with Kai's declining state and culminates with Kai's conversion to Void's side, killing off part of himself while in an unstable state of mind. It gets worse as Kai's loneliness is something that the season is going to cover, leading to his Start of Darkness and in an interview, Takuya Satou reported how difficult he found it to record the episode.
Yuri's fight with Reversed Koutei in Episode 136. She's fighting an opponent she cares for a lot and who she helped rise from being basically an inept beginner to a national champion, and who is now Brainwashed and Crazy to the point of being the exact opposite of his normal self. By the end, she's screaming his name, trying to desperately bring him back to his senses, only to be cruelly defeated by Reverse Daiyusha and reduced to a mindless Reverse fighter. Made even worse if you knew Episode 137's title before, and thus know that she's in a Hopeless Boss Fight.
Gillian and Charlene getting reversed in Episode 137. The entire episode was planned brilliantly to show the audience more in depth about the relationship of Leon and the Twins, and just why the Twins are so loyal to Leon. Their interactions just melted the hearts of the audience only for Reversed Kenji and the Reversed Team SIT to thrash the party. Not only the three of them were separated to fight individually, but Leon was helpless (As he was fighting Kenji) as he sees the Twins get defeated and reversed. Gillian and Charlene were clinging to Leon's memento (a boat origami made by leaves) while trying to reach out to Leon while saying his name (In tears no less, because they had failed him) as they get engulf by the ebony flames...This is definitely one of the most heart wrenching episodes in Vanguard anime history ever.
Aichi's loss against Reversed Kai. While Kai iterates what drove him over the edge, Aichi is on his knees, clutching his chest and screaming while trying to fight off the Reverse. It doesn't get any better after he's saved from converting by Takuto, who says that Aichi is virtually on the brink of death and that if he fights, the Reverse still in him that he's holding back will take over. Even after hearing that, Aichi is still adamant to go after Kai, even though he can't evenstand and has to be forced into passing out just to stop him from putting his life on the line.
Makes it even sadder when you recall that Aichi loved Vanguard so much that he spent the first half of the arc just to establish a club in his school. And t the moment, he can't cardfight at all (Reversed opponent or not). The game that changed him to who he is and got him the friends he has is KILLING him.
Ultra Rare's past, particularly its effect on Kourin. As Takuto houses their memories, stripping them of their past prior to being recruited by him, it's used as an argument against Aichi's fighting, as when Takuto disappears, so will the girls' past memories, and Kourin's since have focused heavily on Aichi, which was her motivation for fighting while Reversed.
An example that is more of a Tear Jerker in hindsight; before leaving to fight the final battle with reversed Kai and Takuto to defeat Link Joker (while on the brink of being reversed) , Misaki makes Aichi promise her that he'll return, alive. Queue season 4, where Aichi has disappeared, and no-one apart from Kai remembers him, not even Misaki and her perfect memory. Her voice is just so heartbreaking when Kai makes her remember.
Episode 183. Despite the massive flak given by the fanbase on the poor writing of the episode and skipping most of the battle (Which was considered the charm of the Vanguard anime), one can't help but to empty their tear glands when Misaki received the Memory Judgment, which caused her to forget Aichi, again. Cue to her standing in Card Capital and staring at Kourin's ribbon, still strapped to her hand, and finally broke down in tears over a reason she can't even remember. The plot has never show mercy to Misaki, but it reaches its peak now.