Having the same problem on my Raspbian (debian) running at a Raspberry Pi 3.
I've actually tried to compile it from source, but I got this:

npm ERR! notsup Unsupported
npm ERR! notsup Not compatible with your operating system or architecture: fsevents@0.3.8
npm ERR! notsup Valid OS: darwin
npm ERR! notsup Valid Arch: any
npm ERR! notsup Actual OS: linux
npm ERR! notsup Actual Arch: arm

I tried using nodejs from apt-get, and tried again using this version (supose to build for ARM):
http://node-arm.herokuapp.com/node_latest_armhf.deb

For compiling, I wrote this:

./scripts/npm.sh install --arch=armhf

Not sure if the npm.sh support the arch parameter

What version of node and npm are you using? I think that error happened when optional dependencies weren't working properly for certain versions of npm 3.

I'm using node v4.2.1

Can you try using node 6.4.0?

The v4.2.1 is the last version I can get at apt-get form Raspbian. At night I will try to get a better version outside package manager.

[EDIT] My node is kind outdated, cause I've checked and I cant even use ES6 javascript, like const or let instead of var.

@jieverson vscode is TypeScript so it should be compiled down, regardless though according to the docs

You'll need Node.JS, at least v5.10.

Warning: npm/npm#12698 prevents us from using npm >= 3.9, be sure to install npm < 3.9.

https://github.com/Microsoft/vscode/wiki/How-to-Contribute#installing-prerequisites

@jieverson sorry for bumping an old thread, but I did get it working on Raspbian on a RPi3 model B.

node --version
v6.4.0
npm --version
v3.8.9

Although node@6.4.0 is not available in the default apt-get repositories, you can still install it with apt-get through these instructions.
You'll then have to manually downgrade npm -- I did this systemwide because I didn't need a super up-to-date version of npm: sudo npm install -g npm@3.8.9. After that, the build just kind of worked to be honest. Either way, good luck!

I really cant get it to work here.

# curl -sL https://deb.nodesource.com/setup_6.x | sudo -E bash

I got some errors trying to install nodesource repo. After the errors, if I try to apt-get nodejs, I still get the v4.

Wow, that really is very strange! I guess you can try using the binaries from node themselves, since they do provide arm-packaged binaries?
For me, the nodesource installer only added (and installed keys for)

deb https://deb.nodesource.com/node_6.x jessie main
deb-src https://deb.nodesource.com/node_6.x jessie main

to my /etc/apt/sources.list.d/nodesource.list file, and didn't touch any mirrors in other apt settings.
I'm running the latest Raspbian image though - your sources seem to imply that you're running Ubuntu trusty?

pi@raspberrypi:~ $ lsb_release -a
No LSB modules are available.
Distributor ID: Raspbian
Description:    Raspbian GNU/Linux 8.0 (jessie)
Release:    8.0
Codename:   jessie