How can I make Google Apps Scripts web apps responsive and mobile friendly?

The way to do this is use the addMetaTag method in your to your .gs file.

 var output = HtmlService.createHtmlOutput('<b>Hello, world!</b>');
 output.addMetaTag('viewport', 'width=device-width, initial-scale=1');

To read more about it look at the documentation here:

https://developers.google.com/apps-script/reference/html/html-output#addmetatagname-content