Yes! It can be done on a web site. I have recently seen this done on a tutorial site, using a privacy setting available per video. I trust this could also be done using an app, but it's possible the access permissons may be different.
Try checking this page for more details: http://vimeo.com/help/faq/managing-your-videos/privacy-settings#what-do-the-different-privacy-settings-do
Pay particular attention to the following point, you can make a video exclusive to your site, but for a premium:
You can select from the following privacy options for video embedding:
Only on sites I choose (PLUS and Pro ONLY)