Search for dates on custom post types

You need to use WP_Query for getting the posts which match your query.

And in WP_Query you need to use the following:

    'meta_query'        => array(
            'meta_key'          => 'date_started', // change it to your custom field meta_key
            'type'              => 'DATE',  // You can also try changing it to TIME or DATETIME if it doesn't work
            'meta_value'        => '2017-05-23', // Replace with your preferred value
            'meta_compare'      => '=',
),

Complete code would look something like:

<?php
$post_query = new WP_Query(
    array(
        'post_type'         => 'post',
        'posts_per_page'    => 1,
        'meta_query'        => array(
            'meta_key'          => 'date_started', // change it to your custom field meta_key
            'type'              => 'DATE',  // You can also try changing it to TIME or DATETIME if it doesn't work
            'meta_value'        => '2017-05-23', // Replace with your preferred value
            'meta_compare'      => '=',
        ),
    )
);

if($post_query->have_posts()) : while($post_query->have_posts()) : $post_query->the_post();

the_title(); // Print the title of found post

endwhile;
endif;
wp_reset_postdata();

?>