notNot the answer you want, but as I understand it canonically the answer is a firm and disputed "maybe". alsoAlso depends on what version of DnD you are running.
From the D&D 3e sourcebook Races of Eberron:
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 wraiths. 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 from the dead, 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 independent 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.
I think they intend it as a plot point for Roleplay and depending on your version their status as "living constructs" means different things.
I would say as a DM if you can't pray without a soul and you can have warforged clerics and paladins then at least Some warforged have souls.
I am sure other have a more thorough knowledge of the fluff than I, but at the same time if iI had a DM that ruled my warforged was soulless iI would definitely use that to munchkin as many advantages as possible.