It's canonically ambiguous.
Like many of the big mysteries in Eberron, this question is left to the DM to decide. This type of intentional moral and cosmological ambiguity is a big part of the theme of the Eberron setting.
The people of Khorvaire consider the question of do warforged have souls? to be unsolved and controversial. The Eberron Races Unearthed Arcana article includes an excellent quote which illustrates this:
“Pierce was built by design, while you were built by accident,” Lakashtai said. “The soul is what matters, not the shape of the vessel.”
“What makes you think he has a soul?” Gerrion said.
“What makes you think you do?”
—Keith Baker, The Shattered Land
The D&D 3rd edition book Races of Eberron, p.8, describes it as ambiguous:
When warforged are used, DMs should be mindful of potential controversies regarding the warforged: Do they have souls?
Even the warforged do not know if they have souls. Some warforged join the Blood of Vol, believing that they will be granted a soul in the afterlife.
What is the argument that warforged do have souls?
A warforged in D&D 5e is defined as living, as a humanoid, and as fully sentient. The method by which they are animated is not well known; it is not by an elemental spirit or some other automatic method. A player can control a warforged character as freely as any other character, and they appear to have free will.
Importantly, they are not given rules exceptions to any spell or effect which affects souls, and are therefore treated as any other humanoid.
What is the argument that warforged do not have souls?
The D&D 5e rules regarding warforged do not define anywhere that they do have souls. However, it must be noted that it is not stated either that humans, elves and such have souls; it is implied only by the existence of raise dead and similar spells, and a paragraph which defines that the souls of creatures pass through the Astral Plane.