Awesome question. Olenna Tyrell (littlefinger was in on it too)
Olenna didn't want her granddaughter marrying a monster. Moreover, she planned on winning over Tommen, and they knew he would be a pushover
Littlefinger likes chaos. And he leveraged the chaos to escort sansa out of the city as the first step of winning the vale/north
If you rewatch the scene, olenna fixed sansas necklace. One of the gemstones went missing. That gemstone was a poison. Olenna put it into Jeffrey's cup
Here's a video I made a long time ago for screenshots if you're bored
View: https://youtu.be/bkf68Xf913A
Extra fun stuff.
Arya has an extended journey with the brotherhood without banners in her Riverlands subplot. On that journey, she meets a woodswitch (greenseer) as opposed to melisandre
The woodswitch, Called the ghost of highheart, has greensight. Visions are often metaphors so as to protect the timeline (melidandres quest to change the future is futile since both the past and the future are effectively already written)
One of the visions she told arya was that she saw a maiden in a castle of snow slaying a giant. The red herring for that is that Robyn arryn says he is a giant and he smashed the snow castle she built at the Vale. Then she slapped him
Although the next book isn't out yet, the giant whom that maiden slays will probably be littlefinger either at the vale during winter or later up north at winterfell. (Littlefinger is jot only a "giant" in terms of the game. Theres an even better reason for the metaphor in the prophecy. So his father started a family on one of the vale peninsulas called the fingers. The littlest one, thus littlefinger. Since his father was from braavos, he used the titan of braavos as the sigil of his house. (The titan was that giant statue that you sail under to get into the braavos harbor). Littlefinger created a sigil of his own, a mockingbird, bur still... his family's sigil might be the clue in the "giant" of thar prophecy
The woodswitch went on to tell arya another prophecy. She said that she dreamed of the same maiden as in the first prophecy (sansa) at a feast with serpents in her hair. The serpents fangs were dripping with venom (in the books, its a hairnet of amethyst, not a necklace). Olenna removes one of them, which was actually poison, and put it in Jeffrey's cup
Another fun theory in the books is that tywin was actually dying before tyrion shot him. In theory, olenna poisoned him as well. She's a bad bitch