While Anne's fine answer assumes that corpses are objects, which has never been confirmed and is an official ruling by Paizo developers (thought it was a rule for D&D 3.5, see the sidebar FAQ) and is subject to table variation, the alternative (covered by this answer) is that corpses are not objects. So, take whichever answer is valid as a rule rule on your table.
Corpses make saving throws normally
A dead character is also unconscious and helpless, as one condition doesn't remove the others unless stated otherwise. Being helpless, they effectivelly have 0 dexterity, so they lose whatever Dex bonus they had on their reflex and take a -5 penalty.
Also note that the rules state that a creature with 0 constitution is dead, but doesn't say that a dead creature loses their constitution. As such, they should be allowed Fortitude saves normally.
They are also most likely prone, and as such, get +4 to AC against ranged attackers, but melee attackers get a +4 bonus to attack them, if that is necessary to land a touch attack.
It is not uncommon, however, to GMs and players to ignore unconscious characters and corpses for area attacks, as that is a ruling that could easily backfire into each other and result in early (and boring) deaths.
The ruling here is a bit different because this is a special case. The body that Vermleks wear is treated as armor, not actually a corpse, or even an exoskeleton that has to be destroyed before the demon can be attacked. Their abilities says:
Flesh Armor (Su)
When a vermlek inhabits a humanoid body (see Inhabit Body below), it treats the dead flesh and muscle as armor, gaining a +3 armor bonus to its AC.
Inhabit Body (Su)
A vermlek can crawl into the body of any dead Medium humanoid, consuming and replacing the bulk of the humanoid’s skeleton and internal organs as it does so.
This process takes 1d4 rounds, during which the vermlek is flat-footed. Once the process is complete, the vermlek appears for all practical purposes to be a living but hideously obese version of the previous humanoid—it gains a +8 racial bonus on Disguise checks to appear as a normal humanoid while wearing a dead body in this manner, but does not gain any of the abilities that the dead creature had in life, including natural attacks, unusual movement types, or natural armor.
It loses its own burrow speed while inhabiting a body but gains the ability to wield weapons or wear armor shaped for humanoids (although the armor bonus granted by wearing armor does not stack with the bonus granted by the vermlek’s flesh armor ability).
So, the "body" that is being inhabited works as a suit of armor, as explained under those two abilities, and you don't need to destroy it to attack the demon directly. An area attack will cause damage directly to the demon, and his "armor" can't make a saving throw to resist damage.
If you want to target the body to destroy it, the one making the saving throw will be the demon, not the corpse. And thought the GM may allow Decompose Corpse to affect the armor similarly to an attack that could destroy armor, the one making the saving throw would be the Vermlek.
For all purposes, it can be treated as a suit of armor, and even be sundered normally.