This script adds a trigger to a Google Form to automatically setup submitted email to become an editor of a specific Google document.

inviteyoself-google-doc.gs

/*

Inviteyoself for Google Doc
===========================

Google doc currently records who edited the file if the file is *not* set to `Anyone can edit'.
We cannot track the author of edits in documents that is set to `Anyone with the link can edit'.

This script adds a trigger to a Google Form to automatically setup submitted email to become an
editor of a specific Google document.

Setup
-----

0. Fill in GOOGLE_DOC_ID with the Google document ID you want to share.
1. Create a Google spreadsheet and add a form that asks for users' email.
   Question for the email must be of type "text" and its title should match QUESTION_TITLE,
   which is seen below.

2. In the spreadsheet of the form, open up Script editor using Tools > Scripts editor...
3. Copy-paste the content of this gist and save the Google scripts project
4. In Script editor, click Resources > Current project's triggers to open up trigger setup.
   Set it up like this: http://i.imgur.com/URt03Gg.png
   so that onFormSubmit() can be run on form submit in current spreadsheet.

*/

var QUESTION_TITLE = 'Email',
    GOOGLE_DOC_ID = 'GOOGLE_DOC_ID_HERE',

    file = DriveApp.getFileById(GOOGLE_DOC_ID);

function onFormSubmit(e) {
  file.addEditor(e.namedValues[QUESTION_TITLE][0]);
}