What would have happened if Harry died during the Battle of Hogwarts? What would have happened to Grimmauld Place and his vault of money and his possessions? Who or what would these items go to?

Since the ministry controls Gringotts and Harry Potter's death would have seen Voldemort elevated to the status of a living God it's not implausible that his vaults would be seized by the ministry and his properties taken over.

DOLORES UMBRIDGE: Have you swallowed something funny in there? Become a Mudblood without any of us noticing? Harry Potter died over twenty years ago as
part of that failed coup on the school — he was one of those Dumbledore terrorists we bravely overthrew at the Battle of Hogwarts. Now come along — I don’t know what game you’re playing but you’re upsetting the dementors and entirely ruining Voldemort Day.

Theere was a unpersonification campaign against Harry Potter in the Voldemort and Valor world and making him look like an terrorist so it's unlikely that any Potter artifacts survived the Ministry's iron hand, let alone find it's way to anyone that could be considered friendly or a relative to Harry.

If Voldemort lost, it’d likely be the Dursleys’.

If Harry died in the battle and the Dark Lord was still killed, since his Horcruxes were destroyed and Harry wasn’t the only one who could kill him, presumably Harry’s wealth would go to the Dursleys. Harry almost certainly didn’t have a will, and in the only other case seen of who inherits a childless wizard’s possessions after their death, if Sirius didn’t have a will, his wealth would have gone to his oldest living relative, his cousin Bellatrix.

“Sirius was the very last of the line as his younger brother, Regulus, predeceased him and both were childless. While his will makes it perfectly plain that he wants you to have the house, it is nevertheless possible that some spell or enchantment has been set upon the place to ensure that it cannot be owned by anyone other than a pure-blood.’

A vivid image of the shrieking, spitting portrait of Sirius’s mother that hung in the hall of number twelve, Grimmauld Place flashed into Harry’s mind. ‘I bet there has,’ he said.

‘Quite,’ said Dumbledore. ‘And if such an enchantment exists, then the ownership of the house is most likely to pass to the eldest of Sirius’s living relatives, which would mean his cousin, Bellatrix Lestrange.”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 3 (Will and Won’t)}}

It’s likely Harry’s possessions would have been inherited similarly, unless there was a rule against Muggles inheriting wizard money.

If Voldemort won, it’s unclear what’d happen.

Less clear is what would happen if the Dark Lord won. Presumably the Dark Lord would want to have all of Harry’s wealth seized by the Ministry, but it’s unclear if the Gringotts goblins would be willing to turn over the portion of it they controlled.