It’s not necessary to create a Widget, just enable the Shortcode capability to the default Text widget.
Add this to the child theme’s functions.php
:
add_filter('widget_text', 'do_shortcode');
Then you can proceed to create a Shortcode that will render the form.
See: How do I create shortcodes for my wordpress themes?
The appearance of the Widget in every page will depend on your theme’s implementation of get_sidebar()
and its widgets areas. But much probably yes, it’ll appear everywhere. Does other widgets appear everywhere you want?