I'm also trying to automate sending an Agreement in background mode. I'd like to send it without user intervention any time a certain type of record is created or updated, meeting certain criteria. I'd also like to do this declaratively. So far, it seems this is not possible.

Page 65 of PDF Onkar referred to provides the same information as this document: 
https://helpx.adobe.com/sign/help/adobe_sign_for_salesforce_advanced_config_guide.html#AutomationandBackgroundActions

The relevant passage says:

"Through automatic workflows or triggers, Salesforce can automatically send a document for signature without additional user intervention. You can also set up trigger or workflows to send reminders and to delete, cancel, or update agreements automatically. To do so, you may need to allow users or workflows to send on behalf of another user."

Triggers, yes. But "automatic workflows" is a fib, or is at least an imprecise use of terminology.

To set up automation, you need an Agreement Template record where you configure:

• master object
• a file attachment that acts as a template of the document that will be signed (I prefer using ContentDocument but there are other options)
• a form field template to map signature and other fillable fields on the file attachment (note that the form field template is configured on Adobe Sign management website, not in Salesforce)
• recipients
• merge mappings
• data mappings
• ability to auto send
• ... other goodies

Once you have that Agreement Template, the following document tells us how to call the apex class to create a new Agreement from an Agreement Template:
https://helpx.adobe.com/sign/help/adobe_sign_for_salesforce_data_mapping_guide.html#template

Example, the following path will use the specific Agreement Template (the templateId parameter) to create an Agreement: 
/apex/echosign_dev1__AgreementTemplateProcess?masterid={!Object.Id}&templateId=a011I000006tTeXQAU

This could be done using a button that a user clicks, or automated by using a trigger on the master object. echosign_dev1 apex classes don't have an invocable method, so you can't call them from declarative automation (Process Builder or Flow).

I'll now discuss with my team, to decide whether we want to do this with apex. Alternatively, my organization uses Skuid, we already have a Skuid wizard built for this specific use case, and Skuid will let me url-hack for days. The Agreement won't send if someone creates the record outside of Skuid... but that might be preferable in this use case. 

Have I overlooked another way to do this declaratively?