You can set the sharing for the document to allow anyone to edit without signing in. Help
Set the document to "Anyone who has the link can edit".
Turn off "Editors can change permissions and share".
From Sharing Settings copy the link to share.
In the site use the Embed function and in the by URL section paste in the copied URL.
For a spreadsheet you can set a notification rule so you get a message when it is changed. There is no notification for documents, but there is a file history so you can revert to an earlier version of the file.
Visitors can edit the document by hovering over the document and clicking the icon to open it for editing.
Also you can embed a link to the Document. Link
You can embed a spreadsheet so that no pop-out icon shows.
<iframe height="500" src="https://docs.google.com/spreadsheets/d/ID##########/edit?usp=sharing&embedded=true&rm=demo" width="75%"></iframe>
With this method viewers can edit it directly on the page.
Idea posted by ChiefChippy2 here
It also works for an editable document
<iframe height="500" src="https://docs.google.com/document/d/ID#######/edit?usp=sharing&embedded=true&rm=demo" width="75%"></iframe>
Note that the first click on the document will move to the top of the page and you must scroll back to it and click again. Google is aware of this problem, but they have not been able to fix it yet.