It was left specifically ambiguous with people in world arguing both ways on whether they have no souls.
Races of Eberron page 16:
THE QUESTION OF SOULS
The Treaty of Thronehold gave warforged their freedom, but only after great debate. House Cannith and Thrane argued ardently that warforged were not living creatures because they do not possess souls. Their evidence for this was that warforged cannot become undead by any known method, not even ghosts or shadows. They are immune to energy drain, and no one knows of a warforged soul in Dolurrh, the Realm of the Dead. Breland argued that because warforged can be raised and resurrected, they must have souls. Of course, House Cannith and Thrane countered that no warforged brought back from death told tales of any kind of afterlife.
In the end, the Question of Souls, as that portion of the negotiations came to be known, was left unanswered. Warforged were freed because they could exhibit thought and free will. Today many people continue to think of warforged as creatures without souls, and citizens of Thrane often refer to warforged as “the soulless.”
Since they are specifically subject to raise dead and 3.5 raise dead requires the subject's soul to be "free and willing to return" for raising to work and 3.5 has rules specifically about how bringing things back from the dead involves bringing their soul back my interpretation is that RAW warforged have souls but in world many in Eberron feel they do not.
3.5 PH page 171 "When a living creature dies, its soul departs its body, leaves the Material Plane, travels through the Astral Plane, and goes to abide on the plane where the creature’s deity resides. If the creature did not worship a deity, its soul departs to the plane corresponding to its alignment. Bringing someone back from the dead means retrieving his or her soul and returning it to his or her body."
In the
3.5 Eberron Campaign Setting book where they were introduced they have sentience and free will and the living construct type. While immune to energy drain they are subject to "ability damage, ability drain, and death effects or necromancy effects" unlike normal constructs. They can also be raised and resurrected.
