How can I query on the year part of a complete date in a custom field?

The handy thing about how ACF stores dates as YYYYMMDD is you can treat them like integers and get a similar level of functionality as if you were using “true” dates.

For example, to get all dates after 1st Jan 2011, use > 20110101. Those before June 1st 2012? < 20120601. And for your case, all dates within 2004? >= 20040101 && <= 20041231.

Translated into a meta query:

$year="2004";

$meta_query_args = array(
    array(
        'key'     => 'production_date',
        'value'   => array( $year . '0101', $year . '1231' ),
        'compare' => 'BETWEEN',
        'type'    => 'NUMERIC',
    )
);

Check out the codex for a full explanation of all the arguments.