The Boggart being able to quickly change forms into many different people's fears would have surely been an excellent weapon against the Death Eaters.
So why didn't someone release the Boggart during the fight with Voldemort's Death Eaters to help turn the tide to Hogwarts during the fight at Hogwarts?

Using a Boggart against many people is not a good idea:

From the lesson on boggarts:

‘This means,’ said Professor Lupin, ‘that we have a huge advantage over the Boggart before we begin. Have you spotted it, Harry?’

‘Er – because there are so many of us, it won’t know what shape it should be?’

‘Precisely,’ said Professor Lupin. ‘It’s always best to have company when you’re dealing with a Boggart. He becomes confused. Which should he become, a headless corpse or a flesh-eating slug? I once saw a Boggart make that very mistake – tried to frighten two people at once and turned himself into half a slug. Not remotely frightening.‘

The premise of the question is incorrect. There is no "the boggart" in Hogwarts. There was a boggart four years earlier, but it is clear from Lupin's statement in Chapter Seven of Prisoner of Azkaban that it was merely a coincidence that the boggart had taken up residence in the castle at the time:

"Boggarts like dark, enclosed spaces," said Professor Lupin. "Wardrobes, the gap beneath beds, the cupboards under sinks — I've even met one that had lodged itself in a grandfather clock. This one moved in yesterday afternoon, and I asked the headmaster if the staff would leave it to give my third years some practice.

Furthermore, that particular boggart was actually destroyed in the Defense Against the Dark Arts lesson in the same chapter:

"Riddikulus!" he shouted, and they had a split second's view of Snape in his lacy dress before Neville let out a great "Ha!" of laughter, and the boggart exploded, burst into a thousand tiny wisps of smoke, and was gone.

While there theoretically could be other boggarts, it seems that boggarts in the castle was not a common everyday occurrence. When Harry wanted Lupin to teach him how to repel dementors in Chapter Twelve of Prisoner of Azkaban, Lupin did not immediately suggest using a boggart. Instead he said that he would need to think about how they would practice. If there was always a boggart or two hanging around, Lupin could have suggested this right away:

"Ah yes," said Lupin, when Harry reminded him of his promise at the end of class. "Let me see... how about eight o'clock on Thursday evening? The History of Magic classroom should be large enough.... I'll have to think carefully about how we're going to do this.... We can't bring a real dementor into the castle to practice on...."

Indeed, later in the chapter they end up using a boggart but Lupin makes clear that it was not easy to find one:

"What's that?" said Harry.

"Another boggart," said Lupin, stripping off his cloak. "I've been combing the castle ever since Tuesday, and very luckily, I found this one lurking inside Mr. Filch's filing cabinet.

Thus, at any given time there might not be any boggarts at all in the castle, or if there are they might be hard to find. When the most important battle against Voldemort is about to begin it would be considered a tremendous waste of time/resources to go looking for a boggart. They might not find one, and even if they did manage to find one it would be rather hard to have it only terrorize the Death Eaters and not the Hogwarts Defenders. Moreover, the Death Eaters might dispose of it in very short order, in which case it would not accomplish much.

A much better use of time/resources was to try to defend the castle in other ways. Ways which did not involve combing the castle with the chance of not finding anything. Ways which did not involve setting something loose that could just as easily turn against them. Ways which did not involve using tactics that could possibly be overcome in an instant with a third-year level charm.

A Boggart wouldn’t be very effective.

Releasing a Boggart to attempt to scare the Death Eaters wouldn’t be very effective. The charm to repel a Boggart is simple, it’s not an advanced or difficult one.

“The charm that repels a Boggart is simple, yet it requires force of mind. You see, the thing that really finishes a Boggart is laughter. What you need to do is force it to assume a shape that you find amusing.”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 7 (The Boggart in the Wardrobe)}}

Lupin is very easily able to repel the Boggart, even when it’s transformed into his own worst fear. The majority of Death Eaters would likely be able to do the same, as they’re all adult wizards and most of them would likely have some level of skill at magic.

“Then they saw a silvery white orb hanging in the air in front of Lupin, who said ‘Riddikulus!’ almost lazily.”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 7 (The Boggart in the Wardrobe)}}

The Boggart-repelling charm wasn’t anything exclusive to ‘good’ wizards, so there’s no reason to think the Death Eaters couldn't cast it as effortlessly and effectively as Lupin was able to. Additionally, the battle was crowded, so the Boggart mightn’t even be able to properly turn into any one person’s worst fear.

“Er – because there are so many of us, it won’t know what shape it should be?’

‘Precisely,’ said Professor Lupin, and Hermione put her hand down looking a little disappointed. ‘It’s always best to have company when you’re dealing with a Boggart. He becomes confused. Which should he become, a headless corpse or a flesh-eating slug? I once saw a Boggart make that very mistake – tried to frighten two people at once and turned himself into half a slug. Not remotely frightening.”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 7 (The Boggart in the Wardrobe)}}

However, even if it did, it’d likely be trivially simple for a Death Eater to repel the Boggart.

The majority of Death Eaters appear to be alumni of Hogwarts and we know that the Defence Against the Dark Arts curriculum has included a lesson on dealing with Boggarts since at least the 1920s.

That being the case, a loose Boggart would be trivially simple for them to deal with.

One other option is: no one wanted to watch on something that may scare Death Eater because it must be something horrible. It will scare everyone around including the person who decide to perform this trick.