As the project code currently stands: No
If your goal is to store (only) your own Story Projects on a web-server instead of: in your web-browser's Local Storage; or on your own hard-drive (or equivalent); then you could fork the project's repository and then write code to replace the web-browser Local Storage based Story Project persistence module with code that uses REST to save the same data remotely, in a similar way to how the code for the NW.js based release replaces the same section of code with it's StoryFile persistence module.
If your goal is to either: allow multiple Authors to store their Story Projects on a single web-server; or to allow multiple Authors to work on the same Story Projects on a single web-server; then the task becomes a lot more complex because you will need to (at least) implement:
a. Some form of Author authentication (and possible authorisation) system.
b. A method to uniquely ID each Story Project so that there can be no overlap.
c. Functionality the allows the web-server to notify an Author that others have modified the Story Project, and knows how to update the local copy of the project in such a way that neither the remote nor the local changes are lost.
d. The ability for an Author to externally backup their Story Projects.
e. Licencing that protects the Author's rights to their content.