"Also I don't want the form to ask the user for authentication in order to upload the file."
You certainly can't upload files into a user's Dropbox without that user's permission.
The Saver is the closest to what you want in terms of allowing a user to save a file to Dropbox without having to authorize your app, but you won't get back a link to the file.
The Chooser also does something like this, which is to give the user the option of uploading a file to Dropbox before then choosing it. You could instruct the user to upload a file to Dropbox in the Chooser and then select it. After that, your app can get back a link to the uploaded file. You can try this out yourself in the Chooser demo and see if it meets your needs.