Errors when downloading the ProcessMap Client Application are usually caused by not having anonymous access enabled for the required application files. These application files are located in the folder ProcessMap/Client.
Some environments, like intranets, typically have specific configuration in place to not allow anonymous access anywhere on the site. This could be through a location tag set on the website root or Anonymous Authentication disabled in IIS.
The ClickOnce process that is responsible for downloading and launching ProcessMap makes anonymous requests to the Client folder. If these requests fails ProcessMap cannot be launched and different kinds of authentication/authorization errors occur.
It is therefore important that the Client folder has Anonymous Authentication enabled.
In addition to this the Anonymous user identity must also have read access to this folder on disk. (the Anonymous user identity can be set through IIS, the default identity is IUSR)