I'm not sure about what you think of when you say "streaming".
WebDAV allows file transfers.
It also allows partial file transfers (as specified in the HTTP standard).
Hence if your application wants to transfer media files to the WebDAV server, this is supported!
And all "media streams" I can think of are - at the end - just files.
(Expect data streams through special purpose protocols like RTP, Skype, etc)
Since WebDAV also supports partial reads and writes of files it's also possible to only transfer parts of a large media file (read: part of the stream).
The WebDAV implementation of OwnCloud is "sabre/dav". It's a very complete implementation of the WebDAV standard.
On the other hand you need to use some client-side implementation to access OwnCloud over WebDAV - depending on the quality of this client implementation you should not run into any problems when transfering (parts of) large files...