First, the Slingbox cannot be played on VLC, and as far as I know, it cannot be configured that way. Slingboxes can be played through whichever Slingplayer software is compatible for the Slingbox. Here is a quick article describing which Slingplayer software is supported for the different models of Slingbox:
Second, the Slingbox 350 does need an internet connection to begin streaming, as it needs to first make a connection to our back-end servers. Once streaming, however, it should continue to stream over LAN, even without internet connection (although if you disconnect from the stream, you will need internet to initialize streaming again). Without any kind of wired internet connection in the home, the 350 cannot connect to our servers to begin streaming. This article touches on the matter:
What the article does not mention is that our Slingbox models that are older than the 350 (PRO-HD, SOLO, other Legacy Slingbox models) can be set up so that you do not need an internet connection to connect to and view the Slingbox. You would need to use our old Slingplayer software to set it up, and you would simply choose to not set it up for internet viewing. If you have no internet connection in your home, an older Slingbox model might be a good solution for you.
Hope this has been helpful!