"For future reference, would a microcontroller be able to call out to a web server and retrieve an updated settings file?"

100% yes. In fact, in my indoor grow automation system, I have a microcontroller for each grow tent, and all of them reach out to a central system over HTTPS and retrieve their configuration from there (with a fallback default config if the server can't be reached). I don't pull a file however, the system returns back a JSON string which I parse as the config.