Which Indo-European invader conquered Fertile Crescent empires through their use of the horse-drawn chariot that carried three soldiers?

The answer is the Hittites.

This were Ancient Anatlian people who established an empire centered on Hattusa in north-central Anatolia. The Hittite military made successful use of chariots, and although belonging to the Bronze Age, the Hittites were forunners of the Iron Age, developing the manufacture of iron artifacts from as early as the 18th century BC.