After some digging, I believe I found a solution to this problem.
ElasticPress uses WP_Query to fetch posts for indexing. When building the query, WP_Query compares the post_status argument against the results of the get_post_stati() WordPress function, which returns an array of registered post stati (statuses).
WooCommerce does this, but it does not appear to happen in time when running elasticpress index from the wp-cli. My solution was to hook the ep_wp_cli_pre_index action and call the WooCommerce registration function manually. Using the solution below, my site is currently indexing WooCommerce orders.
add_action( ‘ep_wp_cli_pre_index’, function( $args, $assoc_args ) {
require_once(ABSPATH . ‘wp-content/plugins/woocommerce/includes/class-wc-post-types.php’);
WC_Post_types::register_post_status();
}, 10, 1 );