Apparently as seen from Hagrid's expulsion, the expelled Wizard or Witch's wand is snapped in half. However, we see on several occasions where people simply replace their broken wand with a new one ( Ron, Neville ).
So why can't an expelled person buy a new wand ? Or are they legally prevented from doing so ?
While it is never directly addressed if it is illegal for a wizard who has had their wand snapped, presumably it would be. The reason their wands were snapped was not so they could not use that individual wand, but so they could no longer do magic. Hagrid specifically said he wasn’t allowed to do magic because he was expelled.
“I’m – er – not supposed ter do magic, strictly speakin’. I was allowed ter do a bit ter follow yeh an’ get yer letters to yeh an’ stuff – one o’ the reasons I was so keen ter take on the job –’
‘Why aren’t you supposed to do magic?’ asked Harry.
‘Oh, well – I was at Hogwarts meself but I – er – got expelled, ter tell yeh the truth. In me third year. They snapped me wand in half an’ everything.”
- Harry Potter and the Philosopher's Stone, Chapter 4 (The Keeper of the Keys)
Since the intended effect is to keep expelled wizards from doing magic, it is almost certain that it would not be legal for them to buy another wand, since that would defeat the purpose of snapping the original wand. Wandmakers and other wand sellers may also play a role in ensuring that no wizards who are not allowed wands buy them. Ollivander knew Hagrid was expelled, and questioned him on if his wand was snapped afterwards and whether he used the pieces.
“Rubeus! Rubeus Hagrid! How nice to see you again … Oak, sixteen inches, rather bendy, wasn’t it?’
‘It was, sir, yes,’ said Hagrid.
‘Good wand, that one. But I suppose they snapped it in half when you got expelled?’ said Mr Ollivander, suddenly stern.
‘Er – yes, they did, yes,’ said Hagrid, shuffling his feet. ‘I’ve still got the pieces, though,’ he added brightly.
‘But you don’t use them?’ said Mr Ollivander sharply.
‘Oh, no, sir,’ said Hagrid quickly. Harry noticed he gripped his pink umbrella very tightly as he spoke.
‘Hmmm,’ said Mr Ollivander, giving Hagrid a piercing look.”
- Harry Potter and the Philosopher's Stone, Chapter 5 (Diagon Alley)
Clearly Ollivander does not believe Hagrid should have a wand, and would not sell him one if he had asked. Therefore it is likely that wandmakers know which wizards were expelled and that they are not supposed to sell them wands. In addition, it is almost certainly illegal for expelled wizards to buy a wand, and possibly even for wandmakers and other wand sellers to sell to expelled wizards, since otherwise it would be trivially easy for expelled wizards to get new wands.
Though nothing like this is ever mentioned, it would be logical for the Ministry to give those selling wands a list of expelled wizards who it would be illegal to sell to as an attempt to enforce the law against expelled wizards doing magic. Even with laws preventing expelled wizards from buying new wands, there are still the possibilities of someone else buying one for them, using the broken pieces of their wands like Hagrid did, or stealing a wand. As it is already in essence difficult to enforce the ban, it would be logical for the Ministry to at least prevent expelled wizards enabling themselves to do magic from being as trivial as walking into Ollivander’s and buying a new wand.
In movies and TV shows when an officer is court-martialed and drummed out of the service they break his sword.* I always thought that breaking a wand is a reference to this. It is a ceremony that demonstrates that the person is persona non grata. A disgraced officer can buy another sword, but it won't make him an officer again. A disgraced wizard can buy a new wand, but he will still be disgraced.
