You need to ask for the child term image field. You have only asked for the current page’s query object image, which may be blank depending on what the query object is.
<?php
$term_id = get_queried_object()->term_id;
$taxonomy_name = get_query_var( 'taxonomy' );
$termchildren = get_term_children( $term_id, $taxonomy_name );
// Loop through each child taxonomy term
foreach ( $termchildren as $child ) {
// Get the term object
$term = get_term_by( 'id', $child, $taxonomy_name );
// Get the values we need to output
$url = get_term_link( $child, $taxonomy_name );
$name = $term->name;
// We make the call here to get the image and we pass in the term we want the image for
$image = get_field('category_image', $term );
// Output it all
echo '<div class="col-sm-4 sub-caregory">';
echo '<a href="' . $url . '">' . $name . '</a><div class="sub-caregory-image"><img src="' . esc_url($image['url']) . '" alt="' . esc_attr($image['alt']) . '" /></div>';
echo '</div>';
}
?>