In the organization where I work, we don't use lightning, so unfortunately I can't offer much assistance in that area. However, I did consider implementing something similar in classic mode (but ultimately decided against it). The following resources may be helpful for you:
Check out this Docusign Javascript Template from Docusign Support
By utilizing the template mentioned above, you have the ability to create your own customized JavaScript Docusign button.
The specific variable that transfers contact information to Docusign is the CRLC variable (which can be found in the provided template).
For more details on customizing contacts, refer to this resource from Docusign Support.
In order to map the fields in the JavaScript button, it's essential to have a Contact lookup field on the quote (one for each recipient).
- The employee selects a contact and fills in the lookup field
- The JavaScript button utilizes the relationship field to extract variables from the contact
FirstName~contact_lookup__r.first_name
This process shouldn't be too complicated, as long as you feel comfortable working with templates or adding the CRL variable to your Docusign setup.