RAW is unclear
Originally, simulacrum did not specify a change in creature type (so a sim of a humanoid would still be humanoid); however, it was errata'd to state:
Otherwise, the illusion uses all the statistics of the creature it duplicates, except that it is a construct.
So the real question is: does a construct have a soul? The introduction to the Monster Manual (p. 6) has this to say about constructs:
Constructs are made, not born. Some are programmed by their creators to follow a simple set of instructions, while others are imbued with sentience and capable of independent thought. Golems are the iconic constructs. Many creatures native to the outer plane of Mechanus, such as modrons, are constructs shaped from the raw material of the plane by the will of more powerful creatures.
Even here, it's not clear if constructs have souls. On one hand, some are merely programmed by their creators. On the other hand, some are "imbued with sentience". Neither gives a definitive answer.
Different spells and effects imply different things
Raise dead only works if "the creature's soul is both willing and at liberty to rejoin the body". It explicitly "can't return an undead creature to life", but provides no restriction for constructs. Resurrection the same restrictions.
Nine lives stealer doesn't work on undead or constructs:
If you score a critical hit against a creature that has fewer than 100 hit points, it must succeed on a DC 15 Constitution saving throw or be slain instantly as the sword tears its life force from its body (a construct or an undead is immune).
This may imply that they don't have a life force to tear from their body. The legendary weapon Blackrazor (DMG, p. 216) has the following similar feature:
Devour Soul. Whenever you use it to reduce a creature to 0 hit points, the sword slays the creature and devours its soul, unless it is a construct or an undead.
A banshee's Detect Life cannot sense undead or constructs:
Detect Life. The banshee can magically sense the presence of creatures up to 5 miles away that aren’t undead or constructs.
The banshee's Horrifying Visage does work against constructs (though undead are still immune), so there may be something special about the (lack of) life in a construct.
The first time this came up, I would probably allow a sim to cast magic jar normally, but remind the player that they're entering a gray area. I'd warn them that I reserve the right to re-evaluate my ruling if the sim + magic jar combo becomes problematic in the future.
There's plenty of ammo to rule either way, but the end result is something you and your group will have to discuss.