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.