I know why Daisy met Tom and that was because of her impatience and her affection with Tom's money but I would like to know Tom's side of the story!
TL;DR: The book doesn’t say, so you’re going to have to imagine it for yourself. I looked on fanfiction.net, but no-one there has written a version of this scene.
There are a few clues on which you might build your version of events. In chapter 8:
And all the time something within her was crying for a decision. She wanted her life shaped now, immediately—and the decision must be made by some force—of love, of money, of unquestionable practicality—that was close at hand.
That force took shape in the middle of spring with the arrival of Tom Buchanan. There was a wholesome bulkiness about his person and his position, and Daisy was flattered. Doubtless there was a certain struggle and a certain relief. The letter reached Gatsby while he was still at Oxford.
That is, Tom met Daisy when he visited Louisville in spring 1919. What took him there? Well, in chapter 1 Nick describes the newlyweds as having:
drifted here and there unrestfully wherever people played polo and were rich together
So perhaps Tom went to Louisville to play polo (the polo season starts in April or May). How were they introduced? Well, it’s possible that there was a family connection. Nick says:
Daisy was my second cousin once removed, and I’d known Tom in college.
It’s unlikely, however, that it was Nick himself who introduced them:
“We don’t know each other very well, Nick,” [Daisy] said suddenly. “Even if we are cousins. You didn’t come to my wedding.”
“I wasn’t back from the war.”
It’s more likely that they met at a social event like a ball, race meeting, or polo match. In chapter 4 Jordan Baker describes Daisy’s life in Louisville in 1917–1919:
The largest of the banners and the largest of the lawns belonged to Daisy Fay’s house. She was just eighteen, two years older than me, and by far the most popular of all the young girls in Louisville. She dressed in white, and had a little white roadster, and all day long the telephone rang in her house and excited young officers from Camp Taylor demanded the privilege of monopolizing her that night. […] That was nineteen-seventeen. […]
By the next autumn she was gay again, gay as ever. She had a debut after the Armistice, and in February she was presumably engaged to a man from New Orleans. In June she married Tom Buchanan of Chicago, with more pomp and circumstance than Louisville ever knew before.
A ‘debut’ was a social event where rich young women were put on display with a view to marriage. I interpret ‘presumably engaged’ as meaning that the couple were not engaged, but behaved as if they were.
As for Tom’s side of the story: he’s described by Nick as ‘arrogant’, ‘aggressive’ and possessed of ‘paternal contempt’, so I doubt that any tender feelings were involved. He was the kind of man to have taken ‘the most popular’ young woman in Louisville as a challenge, or as a prize to be competed for.