Could one of Voldemort's Death Eaters have killed or harmed Harry at Privet Drive?

In Order of the Phoenix, Dumbledore explains to Harry the protection that the Dursleys' home provides:

While you can still call home the place where your mother's blood dwells, there you cannot be touched or harmed by Voldemort. He shed her blood, but it lives on in you and her sister. Her blood became your refuge. You need return there only once a year, but as long as you can still call it home, there he cannot hurt you. Your aunt knows this. I explained what I had done in the letter I left, with you, on her doorstep. She knows that allowing you houseroom may well have kept you alive for the past fifteen years.

Dumbledore says that he cannot be "touched or harmed by Voldemort". Does this mean that he could have sent a Death Eater to just bust down the door and kill Harry, assuming that Voldemort would allow them to? And even if he didn't want them to kill him, could they have harmed him in some other way (e.g. the Cruciatus Curse)?

Answer

No. Harry was protected from Death Eaters and Voldemort's allies, not just Voldemort himself.

Dumbledore explicitly states that when devising Harry's protection at Privet Drive that he was intending to protect Harry from the Death Eaters as well as from the Dark Lord himself.

"My answer is that my priority was to keep you alive. You were in more danger than perhaps anyone but I realised. Voldemort had been vanquished hours before, but his supporters - and many of them are almost as terrible as he - were still at large, angry, desperate and violent."
_{(Order of the Phoenix, Chapter 37, The Lost Prophecy).}

Dumbledore therefore wanted to protect Harry from Voldemort when he managed to regain a body at some point in the distant future. But he also wanted to protect Harry in the short-term from the wrath of Voldemort's supporters. He subsequently cast a protective charm that would defend Harry from any who would wish him harm.

We do additionally have some insight into the perceived impregnability of Privet Drive from key figures from within the Death Eaters. Snape describes it as a "place of safety".

"My Lord, the Order of the Phoenix intends to move Harry Potter from his current place of safety on Saturday next, at nightfall."
_{(Deathly Hallows, Chapter 1, The Dark Lord Ascending)}

And Voldemort says "not even I can touch him" at Privet Drive.

“But how to get at Harry Potter? For he has been better protected than I think even he knows, protected in ways devised by Dumbledore long ago, when it fell to him to arrange the boy’s future. Dumbledore invoked an ancient magic, to ensure the boy’s protection as long as he is in his relations’ care. Not even I can touch him there..."
_{(Goblet of Fire, Chapter 33, The Death Eaters)}

Presumably, that means 'everyone up to and including me'. Essentially, it would be a huge security loophole to only protect Harry from one person and one person only (even if that person was Voldemort). I'm sure that Dumbledore's charm covers any and all individuals who may wish to cause Harry harm. One way of keeping Death Eaters out would be to prevent anyone with a Dark Mark from entering the area.

Even if Death Eaters were somehow able to enter Privet Drive under Dumbledore's protective charm I think the above quotes show that Voldemort believes that Privet Drive is a no-go zone for his forces. Perceived security coverage can be just as effective as actual security coverage. Voldemort believes that no Death Eater can enter Privet Drive. And, as RichS says, he had a specific obsession with personally killing Harry himself; he wouldn't have entrusted that task to one of his minions.