Hello Swennet,
There is no “API” calls for that, you’ll need to connect via sockets in PHP to the twitch irc chat (irc.chat.twitch.tv) and use the IRC RFC references to send message to the channel.
For instance, you’ll need your oauth token that you’ll send as a PASS command parameter once connected to the server, then send the NICK yourNickname to complete authentication (nickname must be in lower case).
Once it is done, you can JOIN the channel you want to send the message to, and use PRIVMSG to send the desired message.
For more informations, please read Github Twitch page about API and IRC chat.
Best regards,
Hugo R.