Who was the first to propose in speculative fiction that citizens all wear bombs that could be remotely activated as a measure to keep people obedient to the law? The earliest I can think of was Jack Vance in The Faceless Man (originally serialized in 1971, later published as a novel, The Anome or The Faceless Man). In this work, citizens of the land of Shant wear explosive collars that can be detonated by radio waves transmitted by the anonymous ("faceless") ruler, who travels unknown among his countrymen. Lawbreakers are decapitated by the explosion. Did Vance have some inspiration from a previous work, or was this wholly original?
1955: "A Ticket to Tranai", a novelette by Robert Sheckley, first published in Galaxy Science Fiction, October 1955, available at the Internet Archive. In a way it's the opposite of what you asked for—it's the government officials, not the ordinary citizens, who wear the bombs—but I think it's close enough to have possibly inspired the works you're interested in. (On the other hand, it seems likely to me that each writer who used the idea thought of it independently.)
The viewpoint character, an immigrant from Earth, learns about the Citizen's Booth:
He was kept so busy that he had practically no time to explore further the mores and folkways of Tranai. He did manage to see the Citizen's Booth. This uniquely Tranaian institution was housed in a small building on a quiet back street.
Upon entering, he was confronted by a large board, upon which was listed the names of the present officeholders of Tranai, and their titles. Beside each name was a button. The attendant told Goodman that, by pressing a button, a citizen expressed his disapproval of that official's acts. The pressed button was automatically registered in History Hall and was a permanent mark against the officeholder.
No minors were allowed to press the buttons, of course.
Goodman considered this somewhat ineffectual; but perhaps, he told himself, officials on Tranai were differently motivated from those on Earth.
He learns more about it as he is about to assume the Supreme Presidency of Tranai:
"There must be certain formalities to go through—"
"No formalities," Borg said, his face shining with perspiration. "None whatsoever. All we do is hand over the Presidential Seal; then I'll go down and take my name off the rolls and put yours on."
Goodman looked at Melith. The immigration minister's round face was expressionless.
"All right," Goodman said.
Borg reached for the Presidential Seal, started to remove it from his neck—
It exploded suddenly and violently.
Goodman found himself staring in horror at Borg's red, ruined head. The Supreme President tottered for a moment, then slid to the floor.
Melith took off his jacket and threw it over Borg's head. Goodman backed to a chair and fell into it. His mouth opened, but no words came out.
"It's really a pity," Melith said. "He was so near the end of his term. I warned him against licensing that new spaceport. The citizens won't approve, I told him. But he was sure they would like to have two spaceports. Well, he was wrong."
"Do you mean—I mean—how—what—"
"All government officials," Melith explained, "wear the badge of office, which contains a traditional amount of tessium, an explosive you may have heard of. The charge is radio-controlled from the Citizens Booth. Any citizen has access to the Booth, for the purpose of expressing his disapproval of the government." Melith sighed. "This will go down as a permanent black mark against poor Borg's record."
"You let the people express their disapproval by blowing up officials?" Goodman croaked, appalled.
"It's the only way that means anything," said Melith. "Check and balance. Just as the people are in our hands, so we are in the people's hands."
"And that's why he wanted me to take over his term. Why didn't anyone tell me?"
"You didn't ask," Melith said, with the suspicion of a smile.
The bomb collars that can be detonated remotely go back at least a few years earlier, to The Reefs of Space (1964), by Frederik Pohl and Jack Williamson. The main characters of the first two novels in the Starchild Trilogy are considered "risks" by the totalitarian "Plan of Man," and they have to wear the explosive collars. The collar is a major plot element in the the first book; the protagonist, Steve Ryland has heard that there is someone out on the titular reefs of space that somehow managed to get the explosive collar off without detonating it, and Ryland needs to find him.
Although the date has already been beaten, the Cyberiad (1965) has an interesting implementation of this idea: there is a planet inhabited by such a civilization. The citizens don't actually wear the bombs, but they are built that way (it's an artificial civilization consisting of synthetic lifeforms). They are ruled by an oppressive monarch, and to prevent conspiracies, they have enough fission material built into their bodies with the purpose that if more than a few of them gather together, it will reach criticality and explode. Basically a form of suppression of the freedom of assembly. The whole story is quite tongue-in-cheek, however.