[The Squatter of Amon Rûdh]

Well, I could be flippant and say that Hobbits of the Shire probably bought their coffee in a shop like any normal person, but that wouldn't address the question of how it got there. In the absence of any information about that, other than that coffee could be bought in the Shire, I'd guess that it was grown somewhere with a climate similar to that of the modern-day coffee-producing countries. Since coffee is grown in Kenya and other African countries, and since Harad is the furthest country to the south that appears on maps of Middle-earth, I'd nominate somewhere in unmapped Far Harad as a point of origin for Shire coffee.

Unfortunately that falls down when we consider trade routes. We need one that allows the coffee to get to the Shire, but avoids those places where they don't appear to have discovered it. By sea to Mithlond does that, but it seems a fairly unlikely trading port. The likely route to somewhere like Pelargir requires some neutral party not mentioned in the books who could act as carrier, and also requires the presence of reasonably serviceable roads into Eriador that Tolkien also neglects to mention. Once the coffee has found its way into the western lands it's easier to imagine it being passed from hand to hand until it reached the Shire, as goods often were in pre-industrial times, but that doesn't work because the Hobbits seem so insular and merchants seem a rare sight in their country.

Since getting hold of coffee by trade seems problematic, and since the hobbits seem somehow to have developed a late-Victorian or Edwardian society, perhaps they grew it in hothouses. That would only require that they somehow discovered the bean and laid their hands on its seeds, which requires only very sporadic and unpredictable contact with far-away and relatively hostile peoples. Some Hobbits were in the habit of disappearing into the blue, so maybe one of those travellers began to cultivate coffee beans, but that is to build tenuous speculation on tenuous speculation founded on more tenuous speculation, and argument cannot live by logic alone. I suppose at least I've offered a possibility.