I don't think you can download the subtitles through VLC, but if you have the subtitles file (usually a .srt file) you can import them by following these steps:
1) Open your video file in VLC
2) Right click on the video which is playing, go to “Miscellaneous” then “Preferences”
3) Click > video > Subtitles
4) Insert your subtitles file.
5) Wait for a few seconds.