If an advertisement for a job stuffing envelopes is fraudulent(which i suspect will usually be the case), does that fraud fall under a mail fraud statute? Is that something that the post office is in charge of investigating? Also, if a social network or website of any kind was paid to allow that kind of advertising, would they be obliged to report it to authorities? Also, if a school, such as a university, has bulletin boards around campus where ads can be posted with permission and a date stamp, is the office involved supposed to refuse the ad?
Many such "job offers' are scams of one sort or another, but not all.
Since this uses US terminology, I am assuming US law here.
Such a scam falls under a mail fraud statute in the US if the mails were used to distribute the ad, or if they are used during the course of the scam, for example, to mail envelopes to the victim or to mail checks to the scammer. The fact that the stuffed envelopes are supposedly intended to be mailed does not, of itself, make this mail fraud.
If such a scam falls under the mail fraud statutes the postal inspectors are empowered to investigate, as is the FBI. But neither is required to do so, nor are other law enforcement authorities precluded from doing so. The FBI generally gets involved only if the scam is perceived by them as "major", and I don't know just where they draw that line. A postal inspector might get involved on a complaint. If the scam is basically local, local police would probably handle it. The state authorities of the state where the scammer is located might take it up.
Quite often no authority will take such a scam up until and unless they are pressed to do so by a complaint.
Private entities are usually not required to report apparent crimes unless a specific statute mandates this, although anyone is free to make such reports. A person of business paid to distribute such a scam-ad might be considered a party to the scam if they knew, or if any reasonable person would have suspected, that the ad was part of a fraudulent scheme. That would still not require a report, but making such a report might make it less likely that the website or network would be prosecuted as part of the scam, and so such a business might be more likely to report if they feared such prosecution, or bad publicity. Much the same would apply to a campus office. How much checking such an office is expect to do would be a matter of school policy -- the law would not require them to check.