Storing files on a server and accessing them using ESP8266

I am making an alarm clock that plays a .wav file when it goes off. I could use an SD card to store and access the file but it is more expensive and annoying to change the file. I was wondering if I could use a wifi Arduino, specifically the ESP8266, to access files I save on a web server.

The user would upload their file to the server and then when the alarm clock goes off it will access the file and play it. I am basically using the web server as an SD card.

Is this possible and how would I go about doing this?

Sounds possible.

An ESP has some buildin flash (4Mb SPIFFS).
That flash can be updated wirelessly (OTA).
No experience with audio+SPIFFS, but it should be enough to store a small wav. file.
Leo..