You need to create a Google UI object that you can embed in your site - you'll do this by creating a Script that you will deploy as a web app. (Refer to Google Apps Script - Web Apps.) That object will contain a button that will invoke your existing script function to send the email.
Publish your web app to execute as you. For your specific situation, you will also want to have access to the app limited to you. That will mean that the UI Blob you create will not function for the general public. This is very incomplete, but produces a blob with a single button that will trigger your script:
function doGet() {
var app = UiApp.createApplication();
var button = app.createButton('Send Schedule Change Email');
app.add(button);
var handler = app.createServerHandler('myClickHandler');
button.addClickHandler(handler);
return app;
}
function myClickHandler(e) {
var app = UiApp.getActiveApplication();
// Call your library function, e.g.
TeamSpreadsheet.sendScheduleChanged();
var label = app.createLabel('Message Sent')
.setId('statusLabel')
.setVisible(false);
label.setVisible(true);
app.close();
return app;
}
Your web app will need to have access to your existing script, which I assume is embedded in your spreadsheet. This is accomplished by first saving a version of your existing script (File - Manage Versions), then adding it as a library to your new web app script (Resources - Manage Libraries). Read more about libraries here. Since your web app will run as you, you can keep your spreadsheet private.
Caveats
The utility script in your library will need to open your spreadsheet in a way that will work from outside of the sheet; use SpreadsheetApp.openById(), rather than SpreadsheetApp.getActiveSpreadsheet(). Unfortunately, openById pukes when you use it to open the host spreadsheet, so you'll want something like this:
var ss = null;
try {
// This works for scripts running in the host spreadsheet
ss = SpreadsheetApp.getActiveSpreadsheet();
} catch(err) {
try {
// This works for web apps
ss = SpreadsheetApp.openById("SPREADSHEET ID");
} catch(err) {
Logger.log("Could not open source spreadsheet.");
// Well, then, not much sense carrying on, is there?
return;
}
}
SpreadsheetApp.setActiveSpreadsheet(ss);
...
In fact, watch for any reliance on calls to get info from "active" objects. You may need to do some gymnastics to get around them.
I've found debugging this type of arrangement to be painful, since the app-script debugger often reports "Server Errors" when you try to trace into a library routine.
I hope that helps gets you started!