<?php if(strlen(get_post_meta($post->ID, 'recent_action', true)) !== 0){ ?>
<a class="button" href="https://wordpress.stackexchange.com/questions/107002/<?php echo get_post_meta($post->ID,"recent_action", true); ?>">Recent Action Link</a>
<?php } elseif(strlen(get_post_meta($post->ID, 'related_link', true)) !== 0){ ?>
<a class="button" href="https://wordpress.stackexchange.com/questions/107002/<?php echo get_post_meta($post->ID,"related_link", true); ?>">Related Link</a>
<?php } elseif(strlen(get_post_meta($post->ID, 'imp_docs_link', true)) !== 0){ ?>
<a class="button" href="https://wordpress.stackexchange.com/questions/107002/<?php echo get_post_meta($post->ID,"imp_docs_link", true); ?>">Important Documents Link</a>
<?php } ?>
or better yet (situation depends on increased reliability):
<?php if(!empty(get_post_meta($post->ID, 'recent_action', true))){ ?>
<a class="button" href="https://wordpress.stackexchange.com/questions/107002/<?php echo get_post_meta($post->ID,"recent_action", true); ?>">Recent Action Link</a>
<?php } elseif(!empty(get_post_meta($post->ID, 'related_link', true))){ ?>
<a class="button" href="https://wordpress.stackexchange.com/questions/107002/<?php echo get_post_meta($post->ID,"related_link", true); ?>">Related Link</a>
<?php } elseif(!empty(get_post_meta($post->ID, 'imp_docs_link', true))){ ?>
<a class="button" href="https://wordpress.stackexchange.com/questions/107002/<?php echo get_post_meta($post->ID,"imp_docs_link", true); ?>">Important Documents Link</a>
<?php } ?>
Using strlen() if you are certain the return will be a string. Use empty() for all other situations.