Yes it is possible, but it's not a trivial process and any solution will require knowledge of web-based programming.
The issues you would need to overcome are:
1. The ability to uniquely identify each end-user of your system.
There are many ways this can be done but the easiest (in my opinion) is to host your story HTML file on a web-server that requires the end-user to login to access your story, this solves the issue of uniquely identity as well as the next issue of secure communications.
2. The ability to securely communicate with your data collection server.
There are many ways to do this as well but if you follow my suggestion in point 1 then you can use the same web-server to store your end-user's data, and you get secure communication (via HTTPS) thrown in for free.
3. The ability to store end-user data on a server.
Again there are many ways this can be done, from storing each individual end-user's data within their own plain text file, to using a (Relational or Object) Database of some sort.