There is a way to eliminate the "Upload" button in Webforms, just adding an "attach" button that uploads the file automatically when you push "send", like in the image.
First, it's worth to remember that file will be uploaded when you push form's submit button, if user haven't uploaded it yet.
Second, you could use AutoUpload module:
AutoUpload is a user interface (UI) enhancement that initiates automatic upload of files minimizing the number of clicks required by a user.
Currently, users must select files, then press the "Upload" button. We found users often don't realize a button press is necessary and mistakenly think their image is uploaded when it's not.
I must admit that this module conflicted with my custom modules and themes, but as a code sample it was perfect. It's GPL, so tailoring it to your custom needs is perfectly legal.