> Is it possible to configure both of these web applications to run using the Wildfly Standalone configuration ?
Yes, wildfly is prepared to run any number of applications deployed on it. The limiting factor is computing resources.
Two applications are fine.
> Would I set up each web application (public and cms) as a host?
Yes, you can deploy both of them but only one can be deployed as a root resource (no context name).
> This deployment does not work when both applications are running. The public facing app runs fine however the cms app continues to throw browser errors.
What are the errors ?