No, the saving throw is not rolled at disadvantage.

As you have noticed, the spell does not involve making an attack roll. (SRD p. 102)

Saving Throws
Many spells specify that a target can make a saving throw to avoid some or all of a spell’s effects. The spell specifies the ability that the target uses for the save and what happens on a success or failure. The DC to resist one of your spells equals 8 + your spellcasting ability modifier + your proficiency bonus + any special modifiers.

Attack Rolls
Some spells require the caster to make an attack roll to determine whether the spell effect hits the intended target. Your attack bonus with a spell attack equals your spellcasting ability modifier + your proficiency bonus.

Have your pet Wraith turn the dead creature into a Specter and then kill the Specter

A Wraith can use its action to turn a recently slain humanoid into a Specter:

Create Specter. The wraith targets a humanoid within 10 feet of it that has been dead for no longer than 1 minute and died violently. The target''s spirit rises as a specter in the space of its corpse or in the nearest unoccupied space. The specter is under the wraith's control. The wraith can have no more than seven specters under its control at one time.

So long as the creature died violently (which can be ensured if you are fighting it) then the Wraith can turn it into a Specter.

The specter has this description in its description in the Monster Manual:

When a ghost’s unfinished business is completed, it can rest at last. No such rest or redemption awaits a specter. It is doomed to the Material Plane, its only end the oblivion that comes with the destruction of its soul.

As a result killing the Specter will destroy the soul of the being it was created from, preventing resurrection.

Necrotic Damage could potentially damage the soul enough that it is destroyed on death

In the Player's Handbook (p. 196), necrotic damage is described thus in the "Damage Types" section:

Necrotic. Necrotic damage, dealt by certain undead and a spell such as chill touch, withers matter and even the soul.

As such it is possible that enough necrotic damage would destroy a soul.

Ensure your target is evil and employ a Night Hag assassin

Night Hags have the following feature:

Nightmare Haunting (1/Day). While on the Ethereal Plane, the hag magically touches a sleeping humanoid on the Material Plane. A protection from evil and good spell cast on the target prevents this contact, as does a magic circle. As long as the contact persists, the target has dreadful visions. If these visions last for at least 1 hour, the target gains no benefit from its rest, and its hit point maximum is reduced by 5 (1d10). If this effect reduces the target’s hit point maximum to 0, the target dies, and if the target was evil, its soul is trapped in the hag’s soul bag. The reduction to the target’s hit point maximum lasts until removed by the greater restoration spell or similar magic.

As a result a Night Hag can permanently kill someone if they have trapped its soul. While the soul is trapped no resurrection is possible.

Have a Barghest feast on the corpse

The Barghest's Soul Feeding sidebar (VGtM, p. 123) has the following text:

This feeding takes at least 1 minute, and it destroys the victim’s body. The victim’s soul is trapped in the barghest for 24 hours, after which time it is digested. [...] Once a creature’s soul is digested, however, no mortal magic can return that humanoid to life.

So after 24 hours the soul is 100% digested and the creature can no longer be revived by mortal magic.

During that 24 hours there is a 50% chance the humanoid can be resurrected on each attempt of revival, but only by a spell that works without the complete original body (so Reincarnate, Resurrection or True Resurrection).

Have your pet Lich feed the soul to its phylactery

A lich can sacrifice a soul to its Phylactery, and after 24 hours the soul is destroyed by the Phylactery. The only way to free the soul is to find the Phylactery and cast a 9th level dispel magic on it, within the 24 hour time limit.

Source: Monster Manual entry on a Lich (p. 203):

Soul Sacrifices. [...] A creature imprisoned in the phylactery for 24 hours is consumed and destroyed utterly, whereupon nothing short of divine intervention can restore it to life.

Summon an Avatar of Death to slay the creature

The Deck of Many Things gives the stat block for an Avatar of Death, and contains this sentence about one of its unique properties:

A creature slain by an avatar of death can't be restored to life.