(See the best response by Doohl.)
How would I go about making a button on my website to transmit some kind of text to my BYOND server?
Best response
There's a lot of ways to go about this, but they're all very complicated. As opposed to having a DMcgi interface/middleman I recommend simply using a network socket to send your data directly to the byond server.

I have some code and information that would save you dozens of hours of work: