Is there a way to connect to a site?

Yes, using HttpService. However, you would need your own website to do what you're asking, and I imagine it'd take a fair bit of coding to get it all working.

Instead, I recommend using DataStores (and the DataStoreService). You can still transmit messages to all servers using them. Dedicate a datastore to your messages and figure out if you want to just have one message in storage, or multiple. If just one, that makes it even easier: choose the datastore and key and have all servers read from this key when they start up (so that the message can be shown to players who join the server). (You might want a second key if you want the message to expire after a while; you could store the time that the message was last updated, for instance.) The servers can use OnUpdate to listen for changes to the key containing the message and update what's shown to the players accordingly. You could have a special script that listens for your chat commands and stores your message(s) if you input the proper command. Alternatively, you can edit your datastores directly in Edit mode (if you have the proper permissions enabled).