VideoJS in a Widget?

To display shortcodes in widgets , use a regular text widget, and add this line to your theme’s functions.php file:

// Use shortcodes in text widgets.
add_filter('widget_text', 'do_shortcode');

This tells WordPress to render the shortcodes in widgets.

and if you want to call your shortcode by php you can use do_shortcode() function

// Use shortcode in a PHP file (outside the post editor).
do_shortcode('

');

To get a better understanding of shortcodes check out this tutorial which covers: What shortcodes are, How to use shortcodes, ,How to define your own. and How to use them in widgets.

Hope this helps