I've heard the out-of-universe explanation is they simply didn't have the props and sets for shuttle scenes that early in the series. Indeed, we don't see any Federation shuttles until The Menagerie, and no shuttles launched from the Enterprise until The Galileo Seven, so I assume this is true.
But is there a known in-universe explanation? I don't recall any characters saying the shuttles were inoperable at the time or the planet had some dangerous atmospheric qualities they needed to bypass via transporter or it would take so long for the shuttles to get down there it wouldn't save Sulu's landing party anyway.