Display post from a date range from custom field

The magic parameter ist described at WP_Query under: Custom Field Parameters

You need to add a meta_query array to your $args where you define your range.

In your case the code could look like this:

$range = array(
    '2015-05-01',
    '2015-05-31',
);
$args = array(
    'post_type'      => 'score_post_type',
    'posts_per_page' => 5,
    'meta_key'       => 'score_time',
    'orderby'        => 'meta_value',
    'order'          => 'ASC',
    'meta_query' => array(
        array(
            'key'     => 'score_date',
            'value'   => $range,
            'compare' => 'BETWEEN',
            'type'    => 'date',
        )
    ),
);

NOTE: Make sure that your date and time values are valid SQL date/time values. Your provided format MM/DD/YYYY is bad for sorting and querying. Better: YYYY-MM-DD.