I wrote a website with Google Sites and associated it to a domain I own, let's call it .com. One of my associates is writing a Web app which will run on app..com. How can I build a Google Sites webpage associated to the subdomain? The instructions in Google Sites only explain how to associate a domain to a site or a page, not a subdomain.
You can add a custom subdomain for your Google Site the same way as you can do for a custom domain. Go to your Google Site -> Settings -> Custom domains. Then you'll need to verify you own that subdomain from Webmaster Central or Google Search Console.