You will be needing Google Apps Script for what you are trying to achieve because in general the e-Signature apps expect the files to be uploaded to their system either as a blob or as a binary. Therefore you will need Google Apps Script support to locate the PDF workflow in your gDrive, convert it to a blob or binary (depending on which type is required by the REST API service that you intend to use), make a HTTP POST call to their API endpoint, upload the file and eSign the document.
Some eSignature REST API services has a feature to email the signed document, but some don’t. If yours have this feature, then you can send thru their Mail Service, if they don’t have then you need to make a secondary HTTP POST call to their API endpoint to download the signed document and send the file via Gmail API.
Vendors which have Email Service for the signed document, generally needs a subscription plan upgrade to use that feature and depending on the vendor’s prices, it might be a bit costly depending on how many documents you sign and send per month. Nevertheless, you always have the option to download the signed document from your own container and email it either manually or via Google Apps Script.
For example; DocuSign is a bit pricey service. If I remember correctly, a 100 signing process was around $1,200 per month with the very basic plan.