In 5th ed, there are a few mentions of souls, but there's no really comprehensive and mechanically-consistent metaphysics established. "Soul" isn't like "attunement," where it's mechanically defined whether or not something has it, and if not, how to get it. So regardless of anything else said here, it's going to have to be adjudicated on a per-game basis by the DM.
This answer's goal is to simply inform that call as much as possible. Also, in that light I'm interested in souls as a mechanical element of D&D, and not in any larger philosophical sense. Since I'm a many-year veteran of D&D, I'll also do my best not to just assume things true in previous editions are true here as well -- but I'm less worried about that.
We know a little bit about souls in 5th Ed based on descriptions of abilities and monsters that interact with them. In Volo's Guide to Monsters, part of the Barghest's entry reads,
A barghest can feed on the corpse of a humanoid that it killed that has been dead for less than 10 minutes, devouring both flesh and soul in doing so... The
victim's soul is trapped in the barghest for 24 hours, after which time it is digested. While a humanoid's soul is trapped in a barghest, any form of revival that could work has only a 50 percent chance of doing so, freeing the soul from the barghest if it is successful. Once a creature's soul is digested, however, no mortal magic can return that humanoid to life.
This, along with other elements of the Barghest's description that explain its mission is to deprive the goblin god Maglubiyet of troops in his eternal war in the outer planes, pretty explicitly lays out what souls are for in D&D 5th:
Souls are necessary for "mortal" magical resurrection of a creature and
Souls travel between the Material and Outer planes when the creature dies.
We also learn that souls exist physically in the Outer Planes, because they are engaged in literal battle.
Unfortunately the Simulacrum entry doesn't explicitly say that the creature can't be raised, which would be a big clue. We might interpret the total disregard the spell seems to have for the persona and lifespan of the simulacrum that way -- and also by describing it as 'the illusion' early on -- but it's not lain out in a definitive way. I'd call it a reasonable inference, at best.
Luckily, in the spell Speak with Dead, we learn that there is a separate force other than a soul that can animate creatures (which we could maybe assume from the existence of mindless undead, but here it's explicit). The description says,
This spell doesn’t return the creature’s soul to its body, only its animating spirit. Thus, the corpse can’t learn new information, doesn’t comprehend anything that has happened since it died, and can’t speculate about future events.
This is immediately helpful because it sounds pretty close to how the simulacrum works. Compare:
The simulacrum lacks the ability to learn or become more powerful, so it never increases its level or other abilities, nor can it regain expended spell slots.
So it sounds pretty reasonable to say a simulacrum does not have a soul, but instead an animating spirit -- something reinforced by its reasonably inferred lack of existence after bodily destruction.