It appears so.
Searching the Monster Manual, we can see several mentions of souls. While not all are listed below, I didn't find any contradictions among the other undead in the book. (Emphasis mine in all quotes)
From the ghost entry (MM p.147):
A ghost is the soul of a once-living creature, bound to haunt a
specific location, creature, or object that held significance to it in
From the lich entry (MM p. 203), while the lich itself doesn't have a soul contained within the body, its soul still holds it to the world via the phylactery:
A lich is created by an arcane ritual that traps the wizard's soul
within a phylactery.
Revenant (MM p. 259) states:
Hunger for Revenge. A revenant has only one year to exact revenge.
When its adversary dies, or if the revenant fails to kill its
adversary before its time runs out, it crumbles to dust and its soul
fades into the afterlife.
The least promising mention is of normal skeletons (MM p. 272), which mentions spirits:
An animated skeleton retains no connection to its past, although
resurrecting a skeleton restores it body and soul, banishing the
hateful undead spirit that empowers it.
It however appears the terms spirit and soul can be used interchangeably, as we can see from the Crawling Claw (MM p. 44) there can only be one spirit per creature, which can pass on.
If a dead murderer's spirit already manifests as another undead
creature, if the murderer is raised from death, or if the spirit has
long passed on to another plane, the ritual fails.
So it appears that even non-intelligent undead possess a soul.
As for using the Magic Jar spell, there are a couple problems, as seen below (emphasis mine).
If the container is destroyed or the spell ends, your soul immediately
returns to your body. If your body is more than 100 feet away from you
or if your body is dead when you attempt to return to it, you die
While it seems an undead can cast Magic Jar and transfer their soul into it, it's an irreversible process.