It’s unlikely that they performed the same level of examination as they did with Dumbledore, but they certainly knew what the will said.
I am not a lawyer, etc.
Sirius was no longer believed to be a Death Eater
After he was murdered by his cousin and recent Azkaban escapee on Ministry property, they began to come to their senses and realise that Sirius was innocent. As Fudge explains to the Muggle Prime Minister:
“Is Serious Black with – er – He Who Must Not Be Named?”
“Black? Black?” said Fudge distractedly, turning his bowler rapidly in his fingers. “Sirius Black, you mean? Merlin’s beard, no. Black’s dead. Turns out we were – er – mistaken about Black. He was innocent after all. And he wasn’t in league with He Who Must Not Be Named either. I mean,” he added defensively, spinning the bowler hat still faster, “all the evidence pointed – we had more than fifty eye-witnesses – but anyway, as I say, he’s dead. Murdered, as a matter of fact. On Ministry of Magic premises. There’s going to be an inquiry, actually…”
— Half-Blood Prince, chapter 1 (The Other Minister)
So they can’t scrutinise Sirius’s will because he’s a convicted Death Eater.
Dumbledore visits Harry on 12th July (according to analysis by the HP Lexicon). This is when we first hear of Sirius’s will:
“I must tell you that Sirius’s will was discovered a week ago and that he left you everything he owned.”
— Half-Blood Prince, chapter 3 (Will and Won't)
This means it was found in early July, and Sirius was killed in mid-June. I think it fairly likely that Dumbledore convinced the Ministry of Sirius’s innocence before his will was found, so that problem goes away. Any attempt to drum up charges on which to search it would look desperate and silly.
The will is probably on public record anyway
Under UK law, your will enters the public record after completing probate (the first step in resolving a deceased person’s estate). It seems reasonable to assume the Ministry of Magic have similar laws.
This means that if the Death Eaters want to find the headquarters of the Order via Sirius’s will, they can just obtain a copy from the Office of Magical Law. No need to do anything suspicious.
Sirius doesn’t need to name Grimmauld Place in his will
Again, I am not a lawyer. But I believe the following would be sufficient for Sirius to bequeath Grimmauld Place without explicitly naming the property:
I give, devise and bequeath all of my worldly possessions and estate, including the rest, residue and remainder thereof, to HARRY POTTER of 4 Privet Drive, Little Whinging, Surrey.
(Based on a real will)
There may have been other legalese to avoid the house passing to another pureblood relative, but that’s the crux of it. If Sirius is savvy, then he doesn’t name Grimmauld Place, so you can’t use his will to find the Order.
When Sirius wrote the will, he was still believed to be a Death Eater, and he’d know that wills were a public record. I’m sure he was quite careful about what he wrote, how he wrote it, and thought about who might be reading it after his death.
That he left everything to Harry isn’t hugely surprising or suspicious – it would be on public record that he was Harry’s godfather.