I want the player to make a change on their server and have that server save it. AND THEN have that change be replicated to every existing server and soon-to-exist server. Is this possible?
You would need to do this using Roblox Datastores by keeping all server components saved inside of one or multiple keys. If you are using one key you can use wiki.roblox.com/index.php?title=API:Class/GlobalDataStore/OnUpdate to detect when the key is added to.
HOWEVER, please note that if you are proposing to have many servers all loading data from other servers and saving it's own, you are bound to go WAAAYYY over the Roblox datastore limits which can be found on this page robloxdev.com/articles/Data-store#Limitations.
To get around this you could use preset servers using wiki.roblox.com/index.php?title=Dynamic_place_creation_and_saving&redirect=no#Limitations, and get the created server ID's to mitigate the number of servers that need to load/save all the data that is coming in.