Vash's status as a Badass Pacifist sparks quite a few, but the most memorable is when he willingly strips down and acts like a dog in order to prevent a hostage-taker from harming anyone. While he could just as easily have shot the gun out of his hand, Vash wants to avoid all violence if possible.
The preview for the episode, which is both heartwarming and tearjerking at the same time:
Kid!Vash: Rem said she came to this ship because someone she loved back on Earth died. I tried to imagine what would happen if Rem died. Suddenly, hot water began to flow from my eyes and didn't stop for a long time. Rem, don't ever leave.
Wolfwood sticking up for Vash after two cowardly villagers called him a wimp.
Wolfwood: Thanks to that "wimp", you two ingrates are still alive!
Lina's grandmother's speech to Wolfwood at the end of the episode.
Grandmother: You don't need to apologize. Mr. Preacher, please take good care of Ericks. Unlike you, he's a bumbling fool of a goody-goody and a sweet, gentle soul. Lina and I have always known somewhere deep in our hearts it was too good to be true. We knew Ericks would probably have to leave us someday, yet we chose not to believe it. Bless him. It doesn't change a fact that we're losing a member of the family. Please, Mr. Wolfwood, don't forget that.
Wolfwood:*(sincerely)* I'm truly sorry. I really am. I'm sorry, Grandma.
Episode 23. Wolfwood's speech gives this troper the sniffles.
Episode 25, when Meryl stops the townspeople from torturing Vash. She repeats Rem's line: "No one has the right to take the life of the others." What makes it heartwarming is that Vash sees her as Rem when she speaks, because he has been holding that ideal in his life.
A minor one, but one episode begins with Milly writing a letter to her family, including her "big-big brother, middle-big brother, little-big brother, big-big sister, middle-big sister, little-big sister" and NUMEROUS nieces and nephews. Meryl chides her for this, since writing to so many family members can keep Milly up too late, but Milly doesn't care, she obviously LOVES her family and is quite happy keeping them updated on her adventures. At the end of the episode, this plus the episode's plot causes Meryl to write to her family, the first she's done in a long time. Awww.
The end of the post-credits scene in the final episode, when Meryl sees Vash come back.
Meryl being unable to confess to Vash before he goes off to fight Knives, knowing it's the last time she might see him alive. She immediately regrets it, but Milly tells her "You can tell him everything when he gets back!"
Double heartwarming since throughout the series, Milly had been one of the only ones to consistently show faith in Vash, electing to believe in him once again.
When Wolfwood gives his meager food to the two hungry kids. If that wasn't good enough on its own, then Vash smiles for real for the first time since the start of the series.
In the manga, the part in Vash and Knive's past where they discovered the horrors of the previous experiments and, after many days of refusing to eat or touch anything from humans, Rem manages to get through to Vash. Especially how Vash (shown surprisingly to be even more rash and violent than Knives in his youth) tried to kill himself with a knife only for Rem to grab it and get stabbed, with him muttering how he won't feel sad about her death and how he won't cry... only to break down sobbing and holding her.