What you can do is go to the MX Toolbox website, and look up the domains. Below I typed in the one that I know uses Google Apps:
As you can see, I typed in “aodocs.com”, and the email service provider is Google Apps. That’s the main piece of Information you need.
Here is also an example of a domain that doesn’t use Google Apps (microsoft.com).
As you can see in the screenshot above, the email service provider is not Google Apps but Microsoft Office.
So, using MX Toolbox makes it super easy to tell what email service provider the domain you search for is using. I think this is the easiest and best way to go.