Dumbledore encouraged his allies to use it.
Dumbledore encouraged those around them to use the Dark Lord’s name. He tells McGonagall that he’d been trying to convince people to call him by his proper name for eleven years.
"My dear Professor, surely a sensible person like yourself can call him by his name? All this 'You-Know-Who' nonsense – for eleven years I have been trying to persuade people to call him by his proper name: Voldemort." Professor McGonagall flinched, but Dumbledore, who was unsticking two sherbet lemons, seemed not to notice. "It all gets so confusing if we keep saying 'You-Know-Who'. I have never seen any reason to be frightened of saying Voldemort’s name."
"I know you haven’t," said Professor McGonagall, sounding half-exasperated, half-admiring. "But you’re different. Everyone knows you’re the only one You-Know – oh, all right, Voldemort – was frightened of.”
- Harry Potter and the Philosopher's Stone, Chapter 1 (The Boy who Lived)
Dumbledore also tells Harry to use the Dark Lord’s name because fear of a name increases fear of the thing itself.
"Sir?" said Harry. "I’ve been thinking … Sir – even if the Stone’s gone, Vol– … I mean, You-Know-Who –"
"Call him Voldemort, Harry. Always use the proper name for things. Fear of a name increases fear of the thing itself."
- Harry Potter and the Philosopher's Stone, Chapter 17 (The Man with Two Faces)
Dumbledore founded and led the Order of the Phoenix.
"Is anyone going to bother telling me what the Order of the Phoenix –?"
"It’s a secret society," said Hermione quickly. "Dumbledore’s in charge, he founded it. It’s the people who fought against You-Know-Who last time."
- Harry Potter and the Order of the Phoenix, Chapter 4 (Number Twelve, Grimmauld Place)
It’s likely he encouraged the Order to use the Dark Lord’s name as well, because Greyback mentioned that the Death Eaters were able to track the Order by tracing the use of the name “Voldemort”.
"You know who used to like using the Dark Lord’s name, Weasley?" growled Greyback. "The Order of the Phoenix. Mean anything to you?"
"Doh."
"Well, they don’t show the Dark Lord proper respect, so the name’s been Tabooed. A few Order members have been tracked that way."
- Harry Potter and the Deathly Hallows, Chapter 23 (Malfoy Manor)
Lupin was in the Order - he was allowed in meetings for Order members only.
Harry made to follow Lupin, but Mrs. Weasley held him back.
"No, Harry, the meeting’s only for members of the Order."
- Harry Potter and the Order of the Phoenix, Chapter 4 (Number Twelve, Grimmauld Place)
Lupin was also in the original Order, since he’s in the picture of the original Order members that Moody brings to show the others.
“Better dead than what happened to them … and that’s Emmeline Vance, you’ve met her, and that there’s Lupin, obviously … Benjy Fenwick, he copped it too, we only ever found bits of him … shift aside there,’ he added, poking the picture, and the little photographic people edged sideways, so that those who were partially obscured could move to the front.”
- Harry Potter and the Order of the Phoenix, Chapter 9 (The Woes of Mrs. Weasley)
As he was in both versions of the Order, Lupin likely learned over the years to use the Dark Lord’s name. Sirius, another member of both the original and reformed Order, also did.
“I mean, all they’ve really said is that the Order’s trying to stop people joining Vol—’
There was a sharp intake of breath from Ron.
‘—demort,’ said Harry firmly. ‘When are you going to start using his name? Sirius and Lupin do.”
- Harry Potter and the Order of the Phoenix, Chapter 6 (The Noble and Most Ancient House of Black)
It seems to take people a while to get used to saying the Dark Lord’s name, so it’s more likely that members of the original Order, who’d been hearing Dumbledore tell them to say it for several years would be willing to use it.