Try this to query a post type to display posts based on a radio button value
<?php
// args
$args = array(
'numberposts' => -1, // Displays ALL post
'post_type' => 'custom_post_type', // Change to reflect the name of your custom post type
'meta_key' => 'custom_field', // Change to reflect the name of your custom field
'meta_value' => 'cats' // Change to reflect the name of the value in your custom field
);
// get results
$the_query = new WP_Query( $args );
// The Loop
?>
<?php if( $the_query->have_posts() ): ?>
<ul>
<?php while ( $the_query->have_posts() ) : $the_query->the_post(); ?>
<a href="<?php the_permalink(); ?>">
<li>
<?php echo get_the_title(); ?>
</li>
</a>
<?php endwhile; ?>
</ul>
<?php endif; ?>
<?php wp_reset_query(); // Restore global post data stomped by the_post(). ?>
