The story HTML you create using the Publish to File option is a standard HTML / CSS / Javascript web-application which means it relies on (and is limited to) the functionality of the web-browser it is being run on.
Web-browsers support three basic methods for sending email:
1. The mailto protocol
When used the web-browser will request the operating system (OS) to open the locally installed email handling application that has been previously configured as the OS's default, and there are many reasons why this request may fail. This method will require the user to interact with said email application.
2. Using the HTTP(S) protocol (directly or via JavaScript Ajax) to communicate with your own web-server.
There will need to be some sort of script on the web-browser to handle the request, and the web-server will need access to an email transfer agent / routing facility (and be pre-configured to use it) like Sendmail.
3. The same as point 2 except communicating with someone else's web-server.
There are numerous on-line companies / services that you can create an (free / paid) account on which will allow you to programatically send email through their site from your static HTML page or web-application, how secure or reliable those sites are I leave up to you (or others) to determine.
I can't recommend a particular service but after a quick search on google some of them are (in no particular order): formspree.io, SendGrid, mailgun, EmailJS, and even via Google Script