Bellatrix‘s wand, like most wizards’, would be like a part of her.
Wizards’ wands are very important to them. The relationship between a wizard and their wand grows stronger as they continue using it over the years. The more they use it, the better it works for them, and the connection between the wizard and their wand increases as they use it and grows stronger over time. Asking them to give up their wand would be like asking them to cut off their arm.
“The best results, however, must always come where there is the strongest affinity between wizard and wand. These connections are complex. An initial attraction, and then a mutual quest for experience, the wand learning from the wizard, the wizard from the wand.” Harry Potter and the Deathly Hallows, Chapter 24 (The Wandmaker)
Bellatrix had (probably) been using the same wand for her entire life, and it had learned from her enough that when Hermione had it, she said it ‘felt like a bit of her’. Bellatrix probably felt the same way about it, and was unwilling to simply give it up, even for the Dark Lord.
“Hermione looked frightened that the wand might sting or bite her as she picked it up.
‘I hate this thing,’ she said in a low voice. ‘I really hate it. It feels all wrong, it doesn’t work properly for me … it’s like a bit of her.” Harry Potter and the Deathly Hallows, Chapter 26 (Gringotts)
Hermione suspected that the wand was Bellatrix’s original wand, and therefore it’d be fairly connected to her and adapted to her style of magic by her having used it exclusively for years.
“But that’s my point!’ said Hermione. ‘This is the wand that tortured Neville’s mum and dad, and who knows how many other people? This is the wand that killed Sirius!” Harry Potter and the Deathly Hallows, Chapter 26 (Gringotts)
She likely was attached to her wand, and didn’t want to give it up. The Dark Lord could relatively easily get a wand, and she probably didn’t want to part with hers. In addition, if she did give it up, it’d mean she would be in the midst of a wizarding war with a less-than-optimal wand. Actually taking part in the war to the best of her ability would be very important to her. She’d want to be front and center in the battles, so she wouldn’t want to handicap herself by having to change wands to one that won’t work as well. She’d know it’d decrease her ability to duel if she could no longer use her own wand, and she definitely wouldn’t want that to happen. She thrived on battle.
“Potter, you cannot win against me!’ she cried.
He could hear her moving to the right, trying to get a clear shot of him. He backed around the statue away from her, crouching behind the centaur’s legs, his head level with the house-elf’s.
‘I was and am the Dark Lord’s most loyal servant. I learned the Dark Arts from him, and I know spells of such power that you, pathetic little boy, can never hope to compete –”
- Harry Potter and the Order of the Phoenix, Chapter 36 (The Only One He Ever Feared)
Since like most wizards she’d likely be attached to her wand, and in her specific case she’d need a reliable one for duels, she’d probably not want to willingly volunteer to give it up when she doesn’t have to. We do see an example of how a wizard feels when forced to part with his wand. When Harry’s wand is broken, he’s extremely upset, and he feels like he lost a part of his magical power.
“The holly and phoenix wand was nearly severed in two. One fragile strand of phoenix feather kept both pieces hanging together. The wood had splintered apart completely. Harry took it into his hands as though it was a living thing that had suffered a terrible injury. He could not think properly: everything was a blur of panic and fear.”
- Harry Potter and the Deathly Hallows, Chapter 17 (Bathilda’s Secret)
He considered losing it worse than many of the other horrible things he’d experienced, including physical injuries, and felt vulnerable without it. Though this was partially because his wand defended him from the Dark Lord of its own accord, from what we see of the relationship between wizards and their wands, most wizards probably feel somewhat the same about their wands.
“Without realising it, he was digging his fingers into his arms as if he were trying to resist physical pain. He had spilled his own blood more times than he could count; he had lost all the bones in his right arm once; this journey had already given him scars to his chest and forearm to join those on his hand and forehead, but never, until this moment, had he felt himself to be fatally weakened, vulnerable and naked, as though the best part of his magical power had been torn from him.”
- Harry Potter and the Deathly Hallows, Chapter 18 (The Life and Lies of Albus Dumbledore)
Bellatrix would know there would be other ways for the Dark Lord to get a wand, so there was no real need for her to volunteer hers but for the possible chance of impressing the Dark Lord (who seemed more to be taunting them all than expecting someone to actually do it), it was unlikely he’d go without a wand. Though she valued the Dark Lord, she still wouldn’t want to just give her wand up, especially since it’d hamper her ability to fight for the Dark Lord and Death Eaters if he took it.
It probably wasn’t that she was afraid she’d remain wandless.
The reason she didn’t want to volunteer her wand probably isn’t because she was afraid that she wouldn’t have any wand at all after giving hers up. The Dark Lord hadn’t said that whoever gave him their wand wouldn’t be allowed to get a replacement, and wizards do sometimes have to get replacement wands. From what the Dark Lord says, it doesn’t seem like she’d be concerned she’d have no wand, and be no better than a Muggle. The only time he implied someone was ‘unworthy’ to have any wand was when he said he would take Lucius’s instead. The only reason why Lucius ‘lost his authority when he lost his wand’ is because of how he lost it - after not getting any offers, the Dark Lord told Lucius it wasn’t necessary for him to have a wand, implying he’s useless.
“No volunteers?’ said Voldemort. ‘Let’s see … Lucius, I see no reason for you to have a wand any more.”
- Harry Potter and the Deathly Hallows, Chapter 1 (The Dark Lord Ascending)
However, the way he asked for volunteers to give him a wand, he didn’t imply they were giving away their right to a wand if they did, just that they were giving him their specific wand. Bellatrix herself eventually has to get another wand because Harry and co. took hers, so her wand was forcibly taken away from her by them, and this doesn’t get in the way of her getting a new wand since she has one at the battle. Even if the Dark Lord was upset at her ‘allowing’ her wand to be taken away, he still (presumably) allowed her to get another one.
“Bellatrix was still fighting too, fifty yards away from Voldemort, and like her master she duelled three at once: Hermione, Ginny and Luna, all battling their hardest, but Bellatrix was equal to them, and Harry’s attention was diverted as a Killing Curse shot so close to Ginny that she missed death by an inch –”
- Harry Potter and the Deathly Hallows, Chapter 36 (The Flaw in the Plan)
Bellatrix, specifically, didn’t have much reason to think the Dark Lord would leave her without a wand at that time. He hadn’t yet mentioned her niece reproducing with a werewolf, and while she fell out of favor a bit after the failure to get the prophecy, she didn’t seem to be being punished in the same way the Malfoy family was. Therefore, her reasoning was probably based on a desire to keep her own wand instead of getting a new one that wouldn’t work as well for her.