It's possible but will require a fair bit of coding, it's not something that can be handled with just a widget by itself.
First your web form would not be a widget, it would be a regular html based form. Upon submit this html form submits the form data to Echosiging via the SendDocument method. (this will take the form values and merges this with a template document, and send it by email to the intended recipient)
The html form would have a field asking for an email, which can be used to set the recipient of the agreement.
I the html form you'd also ask the ssn info, which can be used to set a password in the send document method.
The SendDocument mehtjod will send an email to the set recipient, who will click the link in the email. This will prompt them to enter the password before they can sign.
As for the user from your website, there's 2 main ways of going about this.
if you want the agreement to come from the actual user, then first all your users need to be part of your Echosign account. assuming the form is behind a secure part of the site, you can use the login details for the SenderInfo part of the senddocument method.
You could also have the agreement come from a central email address and capture the form filler via a form field in the main form and merge this data on Echosign document.
I want to add that you will be required to have a Enterprise account because you are looking to embed you website form to the echosign server. This requires to do API coding.
The API coding is simple task but might be a step hill for you to do this on you own. Considering hiring a developer to accomplish this. At the end, you will reach exactly what you are looking after.