In the 5th published legend, the only mention of a soul forged by the war is found in the introduction to the race:
"Pierce was built to design, while you have been by accident," Lakashtai said. "The soul is what matters, not the shape of the ship."
"What makes you think that he has a soul?" Said Gerrion.
"What makes you think you do?"
– Keith Baker, The Broken Earth (Wayfinder Guide for Eberron 69)
The excerpt seems to look at the fact that it's not clear if the warriors have a soul. Nowhere is the issue much less addressed.
In the 2015 version of Eberron Races 5th Unearthed Arcana came out and said this explicitly:
Although they are constructions, they have much in common with living things, including emotions and social bonds, and maybe even souls. (3)
It should be noted, however, that this formulation has not been incorporated into the final published version.
In the 3rd edition, the question was explicitly debated in the universe and did not lead to a conclusion:
At the end, the question of souls, as this part of the negotiations has been known, has remained unanswered. Warforged were released because they could display independent thinking and free will. Today, many people still think of the forge of war as soulless creatures, and the citizens of Thrane often refer to this forge as "those without a soul." (Eberron Races 16)
And explicitly mentioned as an indefinite point on the race:
When fighters are used, DMs need to be aware of potential controversies regarding fighters: Do they have souls? (Eberron Races 8)
It seems that since this race exists, this ambiguity has been deliberately emphasized and left in the course of the race.
Mechanically, they do it efficiently
The only thing that is clear is that the warriors are alive and are affected as if it was something alive by spells and other effects:
Although they are formed from stone and steel, they are living humanoids. Rest, the magic of healing, and the technique of medicine all provide the same benefits to warriors as they do to other humanoids. (Wayfinder Guide for Eberron 69)
Mechanically, this actually means that warriors should be affected by healing magic, including resurrection spells, which would require a soul. So, mechanically, the question does not matter.
No traditional reason is invoked for this, and in fact, the ability to resurrect is canonically one of the arguments that justifies the fact that they could have a soul although it also has a counter:
Breland argued that, as warriors can be resurrected, they must have a soul. Of course, House Cannith and Thrane retorted that no reported weapon of war of death told stories of any kind after death. (Eberron Races 16)
But, as traditions leave an ambiguous answer, your deputy minister can decide the specifics of the flavor of why things work the way they do.