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If memory serves, one of the illustrations in the Backgrounds chapter (I think next to the "knight" variant of the "noble" background, but I could be mistaken) is, in fact, a samurai.
The illustration is actually by the soldier background, however this could indeed be the best way to implement a samurai mechanically in 5e.
Basically, give them History and Intimidation proficiency, proficiency in Calligraphy supplies or similar artistic toolset, some appropriate gear and a feature that reflects their code of honour and also the respect and privilege they gain from that.
You are even allowed to create a custom background in their Adventurers League organised play campaign.