TA was clear - there's no way to write, since static file hosting doesn't support dynamic write to begin with.
However, I imagine a service to support your scenario could be written in a standard back-end server language like Go or JS. The challenges involved would be much greater, however -- how to handle concurrency in particular. I suspect one would do better to just run PostgreSQL behind a web API.
That's basically re-inventing the Database but on the client side. We have gone a long way but we are closer to having the server side just as a data store.
However, I imagine a service to support your scenario could be written in a standard back-end server language like Go or JS. The challenges involved would be much greater, however -- how to handle concurrency in particular. I suspect one would do better to just run PostgreSQL behind a web API.