Because the LogMeIn Rescue Calling Card is installed using Microsoft Windows Installer, you can leverage standard installation commands to install it.
Typically, a silent install for an MSI package is done by issuing the command:
msiexec /qn /i <installername>.msi in a script or at the command line. So, for an msi file named "logmeinrescuecallingcard"
msiexec /qn /i logmeinrescuecallingcard.msi installs the Calling Card as well, with all defaults.
If you only want to install the Calling Card but not to run it immediately upon installation, add
RUN=0 to the end of the command string. So, a fully silent install will be:
msiexec /qn /i logmeinrescuecallingcard.msi RUN=0