How to define WordPress plugin shortcode globally for multisite?

I suggest you to create a Network-only plugin that is Active for all the sites in the network and define your shortcodes inside.

To make specific plugin Network-only you need to add Network: True to your plugin definition, just like the example below.

Change:

/*
Plugin Name: My Awesome Plugin
Plugin URI: http://yourplugin.com/
Description: This is my extremely awesome WordPress plugin
Version: 1.0
Author: John Doe
Author URI: http://yourcompany.com
License: GPLv2
*/

to the following:

/*
Plugin Name: My Awesome Plugin
Plugin URI: http://yourplugin.com/
Description: This is my extremely awesome WordPress plugin
Version: 1.0
Author: John Doe
Author URI: http://yourcompany.com
License: GPLv2
Network: True
*/

After this is done make sure you activate your plugin from the Network Admin and the shortcodes defined in your plugin should work on all the sites in the network.