Amazon Kindle Reader App for PC is packaged with NSIS (Nullsoft Scriptable Install System).

Silent install switch is /S and is case sensitive.

Running without switches will install without user interaction, but will launch the program after installation.  Installing silently simply installs and exits the installer.

Updated 2020-11-20, so close.  If running the install as system (deploying with MEMCM), installer puts the start menu shortcuts into [C:\Windows\SysWOW64\config\systemprofile\AppData\Roaming\Microsoft\Windows\Start Menu\Programs\Amazon\Amazon Kindle], also known as not a useful place. 

I've (disappointingly) created a batch file to run the installer, then run the following..

move "c:\Windows\SysWOW64\config\systemprofile\AppData\Roaming\Microsoft\Windows\Start Menu\Programs\Amazon" "%ProgramData%\Microsoft\Windows\Start Menu\Programs\"

Update 2020-11-23, of course I found if this is deployed as available, and a user installs it, even though the deployment type in MEMCM says to install as SYSTEM, it's installing with user context.  Then I found the permissions on the start menu items was wrong, so this is the batch file I came up with.

:: install the application
"KindleForPC-installer-1.30.59056.exe" /S

::in the event it's run with SYSTEM context (appenforce.log), move the start menu things from here to the proper location.
move "c:\Windows\SysWOW64\config\systemprofile\AppData\Roaming\Microsoft\Windows\Start Menu\Programs\Amazon" "%ProgramData%\Microsoft\Windows\Start Menu\Programs\Amazon"

::in the event it's run with USERS context (appenforce.log), reset the permissions, because they're broken.
c:\windows\system32\icacls.exe "%ProgramData%\Microsoft\Windows\Start Menu\Programs\Amazon" /T /Q /C /RESET

:: STEP 3, PROFIT!