In "Hard Puncher" (Chapter 1/Episode 5), Vash is seemingly taken down, complete with blood on the floor. The looks on the mercenaries' faces when he calmly plugs the gun of one with his finger before appearing behind the rest with a jovial smile is absolutely hilarious.
And immediately topped by him nailing each of them twice with a child's dart-gun when he's threatened... before saying "It's okay, shoot if you want - try your luck!" Then he reveals that the thugs used up their ammunition in the initial Blast Out. How did he know?
Vash: I counted.
Also from Hard Puncher, the Nebraska Family demolishes the building Vash is in... only for the large "son" to retract his Rocket Fist and see that Vash wrote a taunt on his hand in the time it was nearby.
When Meryl tries to talk down the citizens of Inepril City with her megaphone (Ch.2/Ep.5). She's insanely loud but everyone just ignores her, and she has no idea until she gets shot at.
Honestly, any time Meryl is onscreen during the Inepril episodes/chapters is comedy gold. Particularly after Vash becomes the town's hero and he quickly ducks behind her and Milly, letting them take the darts fired by a young boy Vash had befriended. Meryl briefly goes Guns AkimboLaughing Mad while Milly tries to snap her out of it.
Whenever Milly gets drunk, like at the beginning of Chapter 4 (Episode 6):
Milly:(7/8 unconscious)I DISAGREE!! I can handle it! I am unique! I have a separate special stomach just for cake and ice cream!
When Milly thinks she's outwitted Vash by slipping a tracker into his food (Ch.4/Ep.20):
(they follow the signal to under the table where Kuroneko-sama is eating the sandwich) Kuroneko-sama: Nyao~! Milly: Oh my god! He transformed into a cat!
Vash: THE DEADLY DODGEBALL HEAD! A simple technique to hold the ball in place with INTENSE SUCTION! Try this at home, kids!
Pity a nasty case of Mood Whiplash happens literally the very next page (30 seconds later).
Chapter 16/Episode 13, where Meryl and Milly walk in on Vash after he gets out of the shower, isn't funny by itself – it's actuallyrather disturbing – but Vash's "Kyaaa!!!" reaction destroys the drama.
The battle with E.G. Mine in both versions (Chapter & Episode 16) is so pathetic it's funny. At least in the anime he got to actually fire his spikes; the manga didn't even give him that little dignity.
Episode 19 / TriMax Chapter 1 — The famous "Genocide Song":
Vash: (singing) Total slaughter... total slaughter... I won't leave a single man alive... la dee da dee daa... genocide... la dee da dee daa... an ocean of blood... let's begin the killing time. Thug 1: He's... singing? Thug 2: What horrible lyrics! Big Bad: Stay right where you are. He's just trying to scare you. Vash: (sighs) I knew it wouldn't work...
Manga
Nightow's omakes are hilarious. So are his gag covers (thankfully preserved in the 2003 Dark Horse release); Wolfwood is invariably paired with a blow-up doll.
Vash and Wolfwood's confrontation with the Vash imposter at the beginning of Maximum (who was cut from the corresponding anime version in Episode 18). When the phony still insists he's the real Vash despite clearly being outmatched by the genuine article, Wolfwood just falls to the ground laughing his ass off as if thinking to himself "is this idiot for real?!"
In one chapter, Wolfwood looked like he was dying, and Vash was all like "DOOOOOOON'T DIIIIIIIEEEEEEE!" and Wolfwood was stating that he was only going to sleep.
Wolfwood talking about how delicious the noodles he's eating are, then pulling a long-suffering face when he hears onlookers yelling about Vash fighting.
The way Gray The Nine-Lives is finally taken out in the manga: He's crushed by a chunk of ceiling Meryl and Milly are sitting on top of.
The manga-only revelation that Meryl and Millie now work for a broadcasting agency.
Anime
Vash's Establishing Character Moment in the very first episode — He reacts to gunmen destroying the bar he's drinking in by leveling his gun at them. Cool, right? Well, a few minutes later we see what really happened… He pulls the trigger several times getting nothing but clicks, opens the cylinder to see he forgot to load it, screams like a little girl, and runs away as the bandits fire at him.
Vash:*(hiding behind the mook's body)* To the potty, sir!
Ruth: Hmm. *(turns around, then immediately turns back and shoots the Mook when he realizes what's going on)*
At the end of that episode, Descartes tosses his big-ass boomerang. Vash leaps over the weapon and sabotages the retraction mechanism so that it will wrap around Descartes and squeeze him into submission.
Vash: (makes a cat smile) Dangerous toys are fun, but you could get hurt!
Descartes:You Bastard!*(screams as his arm gets twisted backwards)*
During the first few episodes, Vash makes this exaggerated bishōnen face whenever he's around a girl he finds attractive. It wouldn't be that funny, except for the fact that it's so ridiculously over the top and comes out of literally nowhere.
The local rich asshole naming the first bodyguard that walks through the door "Vash the Stampede" (in order to scare off law enforcement), not realizing that he's hired the real Vash.
Also, Ladies and Gentlemen, the single most epic Ship Tease of all time:
Vash: (pointing his hand like a gun at the woman he's guarding) I'm known for my bullets never missing their mark! Especially if it's the heart of a beautiful lady. …BANG!! (Meryl suddenly grabs her chest as though she just got shot, and lets out an extremely melodramatic scream)
Vash: AHAHAHAHA! HEY GRANDMA! WHATCHA WAITIN FOR, FILL ER UP! (bartender bonks him over the head with a beer pitcher) Vash: I mean please fill 'er up, Miss, if you wouldn't mind. Barkeep: That's better.
Later in that same episode:
Vash(who's drunk off his gourd): Gimme...gimme back mah gun grandma... (she tosses his gun at him from offscreen; it smacks him in the face with a loud *THWACK!*) Frank: What happened, brother? Vash: I dunno, my head just started to hurt.
Milly: What are you talking about, Mr. Vash the Stampede? Vash: *Beat* I HATE IT WHEN YOU CALL ME BY MY FULL NAME!!!
The opening to Episode 19 when Vash and Wolfwood end up stranded after Wolfwood's bike breaks down. Fortunately, a random passerby happens upon them and offers to give them a lift... which he quickly comes to regret with the two's constant arguing.
Brad's reaction in Episode 20 when Jessica shows up and she glomps Vash instead of him.
The brief flashback where he confesses to her but she insists she's marrying Vash – whom, let the record show, she had not seen since she was ~6. It's animated in a completely different style that just makes it funny.
The end of episode 25, thankfully ending an extremely depressing episode on a funny note.
Meryl: I hate to ask, but how long are you planning to do this?
Wolfwood killing Leonof the Puppetmaster shouldn't be hilarious* (and when Leonof dies in the manga, by his own hand, it's quite sad). But his yell of "YOU DEPRAVED LUNATIC!" and the explosion knocking him off his perch is, if unintentionally, hilarious.
Knives: My leg. My leg, it hurts, why?...Did you shoot me? Did you actually shoot me?! I can't believe this! Did you really shoot me?!?!?
Vash:(Screams and runs away)
Most of the bar scene in Badlands Rumble, especially the part where Vash manages to steal the bullets out of everyone's guns without them noticing, and drunk Meryl.
