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TL;DR: It appears that they don't though it isn't conclusive either way with the evidence we are given.
Looking through the transcript it appears only those present in the room have a vote. However, people can be voted on that are not necessarily present, though I haven't re-watched the scene to find out.
Ser Denys Mallister joined the Watch as a boy and has served loyally longer than any other Ranger.
- That's right.
- Through ten winters he served.
As Commander of the Shadow Tower he kept the wildlings away. We could do no better.
Then they talk about beginning the voting with only those present in the room, before Sam nominates Jon.
If there's no one else we will begin the voting. The triangular tokens count for Ser Alliser Thorne. The square tokens for Ser Denys Mallister.
Therefore, it would appear you have to be in attendance to vote and as I doubt they'd leave Eastwatch and the Shadow Tower empty it appears as though they don't get a vote. Also as candidates are only nominated there and then this also appears to be the case so they can't vote and send a representative. So tough luck if you aren't there apparently.
On a re-read of the transcript I saw this line:
Does anyone wish to speak for candidates before we cast our tokens for the 998th Lord Commander of the Night's Watch?
So it would appear candidates are nominated before the actual voting, with Jon being an exception, so representatives could come down from the other castles to vote for everyone. However, if I've looked correctly this is never commented on. Though as far as I know we never see Ser Alliser Thorne or Ser Denys Mallister names put forward and the scene begins mid meeting so the names might have only just got put forward there and then.
There is also this quote fro Stannis in his conversation with Jon before the vote:
Tonight, the Night's Watch elect a new Lord Commander. Ser Alliser Thorne is going to win.
Though it was common knowledge that Ser Alliser Thorne was going to run and was popular enough to be voted in so it doesn't confirm either way if the candidates are put forward before the vote meeting or during.
Here's the scene for reference: