Python API for Dropbox

There isn't a programmatic way for obtaining an access token for an account. The user must explicitly authorize the app. Apps can store and re-use access tokens though. If your app has done this, and has an access token for the necessary accounts, then it can use the file/folder moving API call to move the shared folder as desired.

Access tokens are sensitives pieces of data though and should be handled securely, since they allow access to Dropbox accounts. So, for a web app, this is reasonably safe, since all of the access tokens would just be stored on the server and you can implement access control as necessary. However, for a client-side app, every instance of the app should only have access tokens for those local users. I.e., one user shouldn't be able to access another user's access token.