"Else just take a simple cheap webcam and do it yourself. A webcam, a rPi and a bit of coding ought to get you what you need easily."
This is the exact question I'm asking.... a good quality, (cheap if possible), webcam that can quickly be setup for this task of sending images to a server for a web-page to display. So far it doesn't look to exist.
Making it easy for the common user to send something to dropbox is fine, I just agree with Mobydisk that there should be some support for more advanced standards, if a user decides to tread there (WebDAV, SFTP, etc...). And as for FTP, it seems the cameras that support it (e.g., Foscam, Hikvision, etc...) don't support SFTP or FTP-S, the all only support vanilla, insecure, FTP....but that's a different issue.
I went into this thinking it could all be setup in no more than a few hours one afternoon; many of these solutions will take many days. (e.g., rPi is an interesting avenue, and may be something I'll jump into, as I've wanted an excuse to play with one, but it won't be "quick")