Sure. The alternative would be to handle the YouTube Uploads on your site but you users must still have YouTube accounts then which again isn’t user friendly. In the case you want you should probably really better think about completely doing the video stack yourself but then again uploading should be your simplest problem. Encoding and having a good player would be a bigger issue i guess.
If that’s a money project you could still look for some „video hosting for business“ service providers that do exactly that i think.
embedly is just a service for creating thumbs for images/videos. That won’t help you so much. You can of course use any storage for videos but you still need to worry about what formats people upload, encode them, have a webplayer that is cross browser/device compatible and so on.