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.

It is canonically undefined

In published 5e lore, the only mention of a warforged soul is in the introduction to the race:

“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 (Wayfinder's Guide to Eberron 69)

The excerpt appears to be leaning into the fact that it is not clear if warforged have souls. Nowhere else is the question brought up much less answered.

In the 2015 version of the Races of Eberron 5e Unearthed Arcana it came out and said this explicitly:

Although they are constructs, they have much in common with living creatures, including emotions and social bonds, and perhaps even souls. (3)

Though it is worth noting that this wording did not make it into the final published version.

In 3rd edition lore, the question was explicitly debated in-universe and came to no conclusion:

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. (Races of Eberron 16)

And explicitly mentioned as an undefined point about the race:

When warforged are used, DMs should be mindful of potential controversies regarding the warforged: Do they have souls? (Races of Eberron 8)

It seems that through the time this race has existed, this ambiguity has purposefully been emphasized and left as part of the race.

Mechanically, they effectively do

The only thing that is clear is that warforged are living and are affected as if they were any other living thing by spells and other effects:

While they’re formed from stone and steel, warforged are living humanoids. Resting, healing magic, and the Medicine skill all provide the same benefits to warforged that they do to other humanoids. (Wayfinder's Guide to Eberron 69)

Mechanically this actually means that warforged should be affected by healing magic including resurrection spells that would require a soul. Thus, mechanically, the question doesn't really matter.

No lore reason is given for this, and in fact, the ability to be resurrected is canonically one of the supporting points for why they might have a soul though it also has a counter:

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. (Races of Eberron 16)

But, as lore leaves the answer ambiguous, your DM can decide the specifics of the flavor of why things work like they do.

They are Humanoids; therefore, they have souls

In Wayfinder's Guide to Eberron, the entry for Warforged says this:

Living Steel and Stone

---

Warforged are formed from a blend of organic and inorganic materials. [...]

While they are formed from stone and steel, warforged are humanoids. Resting, healing magic, and the Medicine skill all provide the same benefits to warforged that they do to other humanoids. [...]

—Warforged, Wayfinder's Guide to Eberron, pg. 69

The bolded passage is important, because it applies especially to spells like Raise Dead:

You return a dead creature you touch to life, provided that it has been dead no longer than 10 days. If the creature's soul is both willing and at liberty to rejoin the body, the creature returns to life with 1 hit point.

—Raise Dead, Player's Handbook, pg. 270

If Warforged did not have Souls, they'd be unable to benefit from a spell like this. Because their description specifically calls out healing, among other things, as providing as much benefit to Warforged as to other Humanoids, they must necessarily fulfill the same qualifications as other Humanoids for any spell that heals a character. Since Raise Dead is one such spell, which requires its targets to have a Soul, Warforged must have souls.

Yes.

The 5th-level cleric spell Raise Dead says:

You return a dead creature you touch to life, provided that it has
been dead no longer than 10 days. If the creature’s soul is both
willing and at liberty to rejoin the body, the creature returns to
life with 1 hit point.