Mac App Store Apps - How can we install similar to using the "jamf policy -id ###" ?

It's probably less than ideal, but couldn't you scope the apps to a static device group and add the computers to that group, thereby triggering the install?

I don't have any Mac App Store apps, but at least for Mobile Device Apps there is a "Distribution Method" drop-down which allows you to either make available in Self Service or install automatically. This might also be worth looking into.

EDIT: I just re-read your post and it looks like you are using VPP. I'm not sure if/how they can be called manually since they are not really policies.

Set the "Distribution Method" to "Install Automatically/Prompt User to Install" and make sure you have sufficient licenses available (see the VPP tab in the app catalog item). This will ensure that the devices in scope have the apps installed (provided the device meets the system requirements for the app).

I am trying to do similar. If only there was a way to use even command line to trigger an App Store download (with VPP). App Store apps need to receive the same functionality as a regular Policy to make them more palatable for a lot of us Mac Admins.

I just came up with this.  I discovered that you can use applescript to tell Self Service to open one of the self service URLs.  Here is a basic script that seems to work. It does have the side effect of opening self service.  

#!/bin/bash

/usr/bin/osascript <<EOF
tell application "Self Service"
open location "${4}"
activate
end tell
EOF