Hi, @Walshy
In short, a server process that runs periodically should check the list of start times you have, and if it's game time, do some further processing, and if it's not game time, don't do anything.
The first thing that came to my mind was Durable Functions.
With Durable Functions, you can use orchestration functions to perform individual statistical operations. In this case, if the orchestration function can determine that it is the game time, I think that the statistical function should be executed.
You can also implement a program that connects to some database from Azure Functions and saves statistics.
Note: Serverless applications generally have limited computing resources, so you should check in advance if Functions is a good fit.