Joffrey was Poisoned
Joffrey was poisoned during his own wedding to Margaery. He asked Tyrion to serve him a drink. The drink turned out to be poisoned.
Tyrion wasn’t a part of it
Tyrion didn’t knew the drink he was serving to Joffrey was already poisoned.
Baelish arranged it with Lady Olena
Baelish wanted to kill Joffrey for his own reasons while Lady Olena didn’t want her daughter Margaery to marry such a monster , the word Olena used to describe Joffrey.
The Poison was attached to Sansa’s necklace
Dontos , the man Sansa spared his life before, Gave her a necklace as an act of gratitude. The Necklace was given to Dontos by Baelish and it had the poison in one of it’s crystals in a solid form.
Dontos was working for Baelish
Dontos was following Baelish’s orders and he knew he is being a part of a plot. This is why Baelish killed him after the mission was done. (See Game of Thrones: Why Cersei hates Tyrion?)
Lady olenna placed the poison in the wine
During the wedding Lady Olenna took the poison from Sansa’s necklace , without the later knowing, then dropped it into Joffrey’s wine.
Sansa had nothing to do with it
Sansa wasn’t a part of the plot nor she knew what was going on. She really believed that the Neckalce was a gift from Dontos because she saved his life. (See Game of Thrones Season 3 Summary)
Game of Thrones: How Olenna Killed Joffrey
How Joffrey was killed? How did King Joffrey die in Game of Thrones?