There are plenty of anime series out there that are much more fun to watch while stoned, and one has to wonder, do the characters in those shows ever smoke weed themselves?
Realistically, the answer is probably no. Drug use is heavily stigmatized in Japan, and the legal consequences for recreational cannabis use can be pretty steep. Because of this, Japanese media doesn't usually depict drug use, and when drugs do impact an anime story line, it's often not a lot of fun.
That said, it's hard not to think of some characters as total potheads, particularly Stoner from Eureka Seven. Yes, that's his actual name. So, while there most likely aren't many characters who definitely, canonically do indulge in laughing grass, there are plenty of anime characters who probably smoke weed, or at least totally would if given the chance.
Mugen of Samurai Champloo has definitely gotten high at least once. He was part of a group that got ripped out of their minds after a field of what looked to be dank nug was set ablaze. While an accidental contact high does not a stoner make, Mugen's aggressively anti-authoritarian attitude fits nicely with the whole weed thing. He's definitely the type to light up just to stick it to someone who told him not to.
Why is everything such a drag for Naruto's Shikamaru Nara? There are a lot of reasons, but one of them is that he's probably smoking way too much weed. Seriously, this guy spends most of his free time spacing out and staring at clouds. You only do that if you're a nephologist, an artist, or a pothead.
The Yamanakas are some of his family's closest friends, and they specialize in growing and selling plants. There's probably a special Yamanaka strain that Ino, Choji, and Shikamaru spent most their adolescence inhaling. Sure, they risked getting murdered by Shikamaru's mom, but like true ninja in training, they always got away with it. As an adult, he and Temari occasionally eat pot brownies after Shikadai goes to bed.
Stoner of Eureka Seven is basically cannabis incarnate. His given name is actually Stoner, and he spends most of his time rambling about his extremely leftist political views. While Stoner is never explicitly shown rolling blunts, he does smoke cigarettes, which could technically contain just about anything.
While just about everyone in Gintama (besides Shinpachi) probably smokes pot occasionally, the most obvious stoner is Taizou Hasegawa, AKA MADAO, which is short for marude damena ossan (roughly, "totally hopeless old guy" in English). After losing his government job, Hasegawa wanders around aimlessly, taking on odd jobs and sleeping in the park. Hasegawa is a dead ringer for the "lazy pothead" stereotype.