Sauron is a Maia, sort of the Middle Earth equivalent of an angel (or fallen angel, in this case)1. So the real question is "can an angel be killed?" The closest we have to a comparison are the balrogs.
Balrogs are usually considered to be Maiar, fallen angels corrupted by Sauron's old boss Morgoth long before the events of Fellowship. And balrogs can be killed, but it's not easy. I can think of three instances where the death of a balrog is explicitly stated:
- In The Silmarillion Ecthelion, an Elf-lord of Gondolin, kills Gothmog, captain of the Balrogs
- Also in The Silmarillion, Glorfindel of Gondolin kills another balrog
- In Fellowship, Gandalf kills the balrog in Moria
These events have two things in common: all of the beings who killed a balrog had once been in Valinor, Middle Earth's heaven, and they all died in the attempt. So it seems as though killing a balrog is possible, but exceptionally difficult; it takes a special kind of person.
By the time of the Last Alliance, I think the only remaining elf who had dwelt in Valinor was Galadriel2 (The Wizards hadn't yet been sent to Middle Earth at the time), and she wasn't going to be fighting in the battle.
So no individual at the battle was capable of killing a lesser Maia, much less Sauron, a very powerful Maia, short of some astonishingly good luck. So could he have been overwhelmed by the entire army?
We have an example of Sauron's army being defeated in battle: shortly after creating the One Ring, Sauron absolutely decimated the elves but was defeated by the military might of the Numenoreans, men who had been blessed by the Valar (the gods of Middle Earth) and distant ancestors of Aragorn. Presumably he fought in this battle as he fought the Last Alliance, so he can at least be subdued if not destroyed.
However, centuries after this event the Numenoreans fell, and lost the blessing of the Valar. By the time of the Last Alliance, Elendil and Isildur are the closest things to true Numenoreans still alive, and it takes both of them to take them down, with nearly-fatal effort. In Fellowship, Elrond (One of the few participants in that battle alive at the time) recounts the story of Sauron's fall:
Gil-Galad died, and Elendil fell, and Narsil broke beneath him; but Sauron himself was overthrown, and Isildur cut the Ring from his hand with the hilt-shard of his father's sword.
There are lots of ways to interpret that account, and Elrond is clearly condensing the story for the sake of convenience, but three things are clear:
- Elendil died battling Sauron
- Sauron was not completely defeated by Elendil3
- Isildur had to cut the ring from Sauron's finger to end the battle
It took two men of Numenor, both of whom were alive when Numenor sank, to subdue Sauron. Their descendants at the time of Lord of the Rings have much-diluted Numenorean blood - I'd say that Aragorn is the only living man who can come even close to matching Isildur or Elendil and, as much as Sauron fears Aragorn, it's doubtful that he alone could match his ancestor's combined feat.
However, it is possible for Sauron to lose his form (about as close to death as a full Maia can get). This happened at least once, during the Second Age. Sauron was on the island of Numenor, and convinced the Numenoreans to invade Valinor. They failed, obviously, and the Valar sunk Numenor in punishment. Sauron was stuck on the island at the time, and although he wasn't utterly destroyed he lost the ability to assume a pleasant-looking form.
The tl;dr of all this is that it's not inconceivable that Sauron could be forced out of his current form by a sufficiently large and powerful force. However, it's unlikely that any such force remained in Middle Earth by the time of the Last Alliance.
However, even if they succeeded Sauron would not be dead; it's important to note that even after the destruction of the Ring Sauron didn't die: in Return of the King, Gandalf says that:
he will be maimed for ever, becoming a mere spirit of malice that gnaws itself in the shadows, but can never again grow or take shape
1 If we're drawing parallels to Judeo-Christian cosmology, it would be better to think of Maiar as a lesser order of angels; strictly speaking the Valar, often considered the "gods" of Middle-Earth, are more accurately classed as angels (With Eru Illuvatar as The One True God), and Valar and Maiar are technically the same kind of creature, just with a different level of power.
2 As Mike Scott reminds me in comments, there's some debate over whether there are two Glorfindels or just one. See my comment here for the background knowledge. So it's possible that there was another elf walking around with the ability to at least kill a lesser Maia. Whether or not he could defeat one or Sauron's strength is purely speculative, because we no longer have any canon support.
3 You can argue over how much Sauron was weakened by his fight with Elendil, and the Jacksonverse argues that it wasn't very much - he had enough fight in him to curb-stomp Isildur for a while - but it's clear that Sauron was still alive and kicking when Elendil was killed. What's not clear is how much effort it took Isildur to cut off the ring; it could have been little, it could have been nearly fatal.