Is it possible to embed a document into a salesforce lightning page (or any web page), so when clients visit that page they can easily sign the document without getting it into the email?

I would suggest looking into DocuSign Powerforms, which are basically online Templates. Upon creating a Powerform you can copy a URL and then add that to a web page or have a button redirect to the URL. I have not seen it done inside Salesforce but definitely on a web page. Here is the DocuSign video where it is best to start....https://support.docusign.com/en/videos/Create-PowerForm