We need to build the .war files in Appeon from our application for an initial at our customer site who has Appeon AEM installed.

Can we build those files offsite and deliver them for installation using AEM and the application server or must they be build on-site on the customer network?

YES you can follow up the package wizard in the Appeon developer to generate portable deployment packages for Web applications. Thus your customer can directly deploy the Web application package to the server without installing Appeon Developer and using it to do Web deployments (perhaps the customer needs to deploy the .war file from the application server console manually).

Please refer to the Appeon Help | Appeon Developer User Guide | Packaging Applications chapter for more details.