Although there isn't a native feature that will update an app or the contents of your website by sending a text message, by leveraging the power of the platform and using readily available third-party tools, this is very possible.
One way is to use a dedicated Twitter account. Set up the Twitter account so that your designated person (Superintendent, PR director, principal, etc.) registers their cell phone with that dedicated Twitter account - only text messages from this phone will post to the website. On the website, set up an announcement that displays the Tweets from this dedicated Twitter account - there are many ways to do this, including some simple JavaScript codes freely available online. With this all set up, the designated person sends a Tweet via text message, which then displays on the Announcements app on the website. To "clear" the announcement, simply Tweet again, sending the HTML code for a space ( ).
I have a demo of this if you're interested in seeing it in action. Email me and I'll be happy to walk you through it.
--Jeff