I would like to use the Email To field in document action from another object that is not Lead Or Contact ID (these objects are specified as only objects to work with automating merge field in “Email To” field in document action article).
If you setup a button to generate and email the document with an email preview then you can populate a merge field in the Email Additional To in the button code and or on document action to populate an email address or multiple that you can merge from some formula of related record for example. That will generate the document and then compose an email and populate the email additional to field on the standard send email page without needing to use a Lead or Contact ID in the email to field (you can leave the Email To field blank in document action).
If you want to have an automated email send in background without user clicking the button to generate document from record, this requires a Contact or Lead ID as explained in the document action article. The Contact and Account objects are included in Force.com licenses as standard objects and are the only standard objects included. So you can leverage them for your base object in document action instead of building a custom one.
You could also consider creating a matching account/contact in background and using a lookup field to relate them. Typically all emailing type functionality across many apps and integrations with email marketing tools requires use of leads or contacts (for tracking email activity, etc.).