Hi @mshah72 !
I do not think it is possible out of the box, but certainly is possible.
Adobe Sign has an API so it
https://secure.na1.adobesign.com/public/docs/restapi/v6
Which seems to be having option to upload a file and specify recipients:
https://secure.na1.adobesign.com/public/docs/restapi/v6#!/agreements/createAgreement
Meaning it could be created pushing data from Airtable via Automation to Adobe (* however I did not check authentication format.
I see that you can also use Make.com and they have the "Create an Agreement" module:

I have not used the Adobe API, so it would require some time dig through docs and test, but you could try to see if it can work out yourself in Make.
If you would be interested to use DocuSign - I have somewhere implementation instructions. Cannot tell though if Adobe or DocuSign would be better for your use case.
I hope this helps!
You can use any eSignature platform that you’d like, and automate it with Make’s advanced automations & integrations,
For example, you can automate the communication between eSignatures.com and Airtable by using Make’s eSignatures integrations along with Make’s Airtable integrations.
But you can choose from any eSignature platform that you would like, and automate it with Make.
If you’ve never used Make before, I’ve assembled a bunch of Make training resources in this thread. For example, here is one of the ways that you could instantly trigger a Make automation from Airtable.
Alternatively, one of the easiest (and cheapest) ways of doing this is to use Fillout’s advanced forms for Airtable.
I show how to do this step-by-step on this Airtable podcast episode:
Fillout also offers hundreds of features that Airtable’s native forms don’t offer, including the ability to update Airtable records using a form, accept payments on forms, display Airtable lookup fields on forms, create new linked records on a form, control access to a form via SSO or email domains or a list of email addresses stored in Airtable, perform math or other live calculations on your forms, collect signatures on a form, create multi-page forms with conditional paths, connect a single form to dozens of external apps simultaneously, add CAPTCHAs to your form, and much more.
Here’s another podcast episode where I show off a few more of the advanced features of Fillout:
Hope this helps!
If you have a budget and you’d like to hire the best Airtable consultant to help you with this or anything else that is Airtable-related, please feel free to contact me through my website: Airtable consultant — ScottWorld