Quick question...which case applies?

1. If you just need a query for today's date being between two pieces of metadata, I'd create a widget running a fairly straight forward WP_Query and outputting the posts. 

2. If you are asking for the user to input a date to search, I'd probably still use a widget but would use the jQuery UI datepicker method for better user interface and data integrity. 

Either way, I wouldn't repurpose the stock search form (which is just a widget anyway).

OK...if it's option 1, try this...
build the widget in your functions file.
    
    class my_events_widget extends WP_Widget {
		//custom properties
		public $base_id = 'my_events';
		public $title = 'My Events Title';
		public $description = "Today’s Events";
		//end
		function __construct(){
			parent::__construct($this->base_id,$this->title,['description'=>$this->description]);
		}
		public function widget($args, $instance) {
			$str = '<div class="widget"><h2>'.$this->description.'</h2><ul>';
			$q = new WP_Query([
				'post_type'=>'events',
				'meta_query'=>[
					'relation'=>'AND',
				[
					'key'=>'start_date',
					'value'=>date('Y-m-d'),
					'type'=>'DATE',
					'compare'=>'<='
					],
				[
					'key'=>'end_date',
					'value'=>date('Y-m-d'),
					'type'=>'DATE',
					'compare'=>'>='
				]]
			]);
			if ($posts = $q->get_posts()) {
				foreach ($posts as $post) {
					$str .= sprintf('<li><a href="%s">%s - %s</a></li>', esc_attr(get_permalink($post)), $post->post_title, get_the_date('M j, Y', $post));
				}
			} else $str .= "<li>Nothing posted currently. Please check back.</li>";
			$str .= '</ul></div>';
			echo $str;
		}
	}
Now register the widget in your theme's functions file.

    function my_custom_widgets() {
      register_widget('my_events_widget');
    }
    add_action('widgets_init','my_custom_widgets');

You can put the widget in a sidebar or use it in a page/post as a template tag using [`the_widget()` function][1].

Might need to tweak the WP_Query a little since I couldn't fully test it out but it should get the job done.


  [1]: https://developer.wordpress.org/reference/functions/the_widget/