A simulacrum is a creature; the rules support it casting wish
"The duplicate is a creature, partially real and formed from ice or
snow, and it can take actions and otherwise be affected as a normal
creature." (p. 276, PHB).
Wish is the mightiest spell a mortal1 creature can cast. (p. 288, PHB)
At first blush, being a creature it can cast the wish spell.
Is the simulacrum alive or dead? The strongest case is for alive ("partially real") but this is open to some question (see repair below). I see partially real as a condition like partially pregnant: you either are, or you aren't.
Once the spell is cast, there it is standing there in front of you, your simulacrum. It's real, and it's really there (in game reality) until dispelled or reduced to zero hit points. It acts as a creature acts.
It has the capacity for emotions (because it can be friendly) and it has life, albeit 1/2 the HP of the caster.
One of the cons to this approach is that when it takes damage, the simulacrum is repaired rather than healed.
If the simulacrum is damaged, you can repair it in an alchemical laboratory, using rare herbs and minerals worth 100 gp per hit point it regains. The simulacrum lasts until it drops to 0 hit points, at which point it reverts to snow and melts instantly. (PHB, p. 276)
So while it dies, or is destroyed, when reduced to 0 HP (similar to most creatures) the simulacrum (like a door or a wall) is repaired if it has taken HP without being destroyed. Most creatures are healed of their damage/wounds. We don't see any rule that a simulacrum could be raised from being dead via the appropriate spell (reverts to snow and melts) as mortal creatures can be so that's a point against the simulacrum having a soul, and being a mortal creature rather than a thing, or an object.
What kind of creature is it?
A simulacrum is classified as a construct. It is made not born. (MM p. 6, Constructs). The simulacrum arises from "snow or ice in quantities sufficient to make a life sized copy of the duplicated creature." (PHB, p. 276)
Constructs are a creature type just as undead or dragon or beast is a creature type. They can be programmed automatons, or can have sentience and independent thought.
Some are programmed by their creators to follow a simple set of instructions, while others are imbued with sentience and capable of independent thought. (MM, p. 6)
Compare the simulacrum to a homunculus (a construct) in terms of relative power, strength, and complexity.
Given the level of spell that this made-not-born creature comes from (7th), and the capacity in the description for having emotions, sentience and the capacity for independent thought, ruling that it is a mortal creature and can thus cast the wish spell is reasonable and looks to have solid rules support.
- @BenBarden points out that being a creature suffices to be able to cast a wish; being a mortal creature or not a mortal creature does not seem to be germane. That strengthens a ruling that a simulacrum can cast the wish spell.
- In support of that point, even though a regular efreeti (a creature, PHB, p. 145) cannot cast the wish spell, an efreeti from an Efreeti Bottle (DMG, p. 167) is still a creature (MM is referred to) can cast three wishes:
The efreeti can cast the wish spell three times for you. It disappears when it grants the final wish or after 1 hour, and the bottle loses its magic.
Ruling against is not as well supported
Choosing to emphasize the made / repaired quality of the simulacrum would support a ruling against it casting the spell. The problem with that ruling is: what other spell can't it cast it if can't cast that particular spell?
1 Mortal creature: in the context of D&D 5e, souls exist as does an actual afterlife. If the in-game meaning of 'mortal' means that a creature has a soul, then a simulacrum is not a mortal creature, but it is still a creature. Given the research @pokep did for the answer on desires and mortal, that quality is not specified (though it could be assumed or inferred).
FWIW, Adventurer's League rules operate in the basis that simulacrum can cast a wish:
You Are You; and So Is He. If a simulacrum you have created casts wish, both you and your simulacrum suffer the stress associated with casting the spell—including the risk of being forever unable to cast wish again. The inability to cast wish extends to any simulacrum you create in the future. (AL, FAQ).