Before attempting to build make sure you have installed and working versions of:
npm
mvn
sbt
vsce
To build use:
npm run package
If successful this will create an apex-assist-x.y.z.vsix file
By default I build on OS X, but the process should work on Linux. You should be able to build on Windows by performing the same actions as the "package" script from package.json, but the scripts are setup to assume a shell is available.
The build process is rather involved due to the different stacks in use.