Most copiers do have this capability now, in most cases when purchasing or leasing a new copier you'll need to purchase some type of app in order for the copier to access your dropbox account.
Depending upon the manufacturer and the service offered you may also have the ability to print documents from Dropbox, just make sure you check with your Imaging Professional first.
With our Ricoh systems you'll need to purchase ICE (Integrated Cloud Environment) for about $240 per year. You'll only be able to scan to a repository/folder, thus if you need to put the file in another folder you would have o log on to dropbox from your smartphone, tablet or PC and then manually move the file to the desired location.
What many uses may not realize, is that since dropbox sets up as a folder on your PC, you can point the copier to scan to the path of your dropbox folder on your PC. You can do this without buying ICE and you can do this with older Ricoh MPF's that are do not support a web browser or Ricoh ICE. I've been using this with an older Ricoh MPC4000 and it's worked perfectly for the last year or so.