Open up your command line interface of choice and navigate to the vSphere Web client scripts directory which is located in the install directory where it was installed. For example:

InstallDir\VMware\Infrastructure\vSphereWebClient\Scripts

Once there execute:

Admin-cmd.bat register <vsphere-web-client-url> <vcenter-server-url> user password
(ex. Admin-cmd.bat register https://MyNewVcenter:9443/vsphere-client https://MyNewVcenter AdminUser AdminPW123)

You will then need to take action on the untrusted certificate prompt:

It will notify you of a successful completion. If you run the “admin-cmd” by itself it will output some help.