Some of those who went to school with the Dark Lord likely knew.
Tom Riddle mentions that he’d been already using the name ‘Lord Voldemort’ with his closest friends (though he likely used the term ‘friends’ loosely, as Dumbledore said he never had or wanted friends). Some people, at least, knew that he was becoming Lord Voldemort.
“Then he waved the wand once, and the letters of his name re-arranged themselves:
I AM LORD VOLDEMORT
‘You see?’ he whispered. ‘It was a name I was already using at Hogwarts, to my most intimate friends only, of course. You think I was going to use my filthy Muggle father’s name for ever? I, in whose veins runs the blood of Salazar Slytherin himself, through my mother’s side?”
- Harry Potter and the Chamber of Secrets, Chapter 17 (The Heir of Slytherin)
Most likely, he’d also have told his followers at the time about his opening the Chamber of Secrets and being the Heir of Slytherin, since, like the name he created for himself, he’d be proud of it and want to boast to those who would be impressed by it as he does to Harry.
Lucius also likely knew that the Dark Lord opened the Chamber.
Lucius Malfoy likely knew that the Dark Lord was the one who opened the Chamber of Secrets, since even his house-elf Dobby knew the Dark Lord was involved in opening it the second time, and presumably Dobby won’t know more about the Dark Lord’s opening the Chamber than Lucius.
“I’ve just got one question, Dobby,’ said Harry, as Dobby pulled on Harry’s sock with shaking hands. ‘You told me all this had nothing to do with He Who Must Not Be Named, remember? Well –’
‘It was a clue, sir,’ said Dobby, his eyes widening, as though this was obvious. ‘Dobby was giving you a clue. The Dark Lord, before he changed his name, could be freely named, you see?”
- Harry Potter and the Chamber of Secrets, Chapter 18 (Dobby’s Reward)
Dobby also seemed to know about the last time the Chamber was opened, since he warns Harry that history will repeat itself, implying he knows something about what happened the last time.
“And now, at Hogwarts, terrible things are to happen, are perhaps happening already, and Dobby cannot let Harry Potter stay here now that history is to repeat itself, now that the Chamber of Secrets is open once more –”
- Harry Potter and the Chamber of Secrets, Chapter 10 (The Rogue Bludger)
He also says he can’t tell Harry who opened it the last time, which implies he’s bound to secrecy by the enchantment on house-elves rather than that he doesn’t know who opened it then.
“Who’s opened it? Who opened it last time?’
‘Dobby can’t, sir, Dobby can’t, Dobby mustn’t tell!’ squealed the elf.”
- Harry Potter and the Chamber of Secrets, Chapter 10 (The Rogue Bludger)
The Dark Lord likely told Lucius about the first time he’d opened the Chamber no later than when he gave him the diary that would reopen it, though possibly even before then.
“When Voldemort discovered that the diary had been mutilated and robbed of all its powers, I am told that his anger was terrible to behold.’
‘But I thought he meant Lucius Malfoy to smuggle it into Hogwarts?’
‘Yes he did, years ago, when he was sure he would be able to create more Horcruxes, but still Lucius was supposed to wait for Voldemort’s say-so, and he never received it, for Voldemort vanished shortly after giving him the diary. No doubt he thought that Lucius would not dare do anything with the Horcrux other than guard it carefully, but he was counting too much upon Lucius’s fear of a master who had been gone for years and whom Lucius believed dead.”
- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)
Though he didn’t know the diary was a Horcrux, Lucius did know it would reopen the Chamber.
“Of course, Lucius did not know what the diary really was. I understand that Voldemort had told him the diary would cause the Chamber of Secrets to reopen, because it was cleverly enchanted.”
- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)
Because of this, and his house-elf Dobby’s evident knowledge of how the Chamber was opened last time, it seems almost certain that Lucius knew that the Dark Lord had opened it the first time.