Is it possible to update the total price on the checkout page in woocommerce from a custom field

Well, you can use a WooCommerce hook( woocommerce_before_calculate_totals )
I will give an example so that it may be easier to understand what you believe.
As far as you know , I think you problem will be fixed

    function calculate_embossing_fee( $cart_object ) {
    if( !WC()->session->__isset( "reload_checkout" )) {
        /* Gift wrap price */
        $additionalPrice = 5;
        foreach ( $cart_object->cart_contents as $key => $value ) {
            if( isset( $value["embossing_fee"] ) ) {
                // Turn $value['data']->price in to $value['data']->get_price()
                $orgPrice = floatval( $value['data']->get_price() );
                $discPrice = $orgPrice + $additionalPrice;
                $value['data']->set_price($discPrice);
            }
        }
    }
}
add_action( 'woocommerce_before_calculate_totals', 'calculate_embossing_fee', 99 );

I will Suggest to you can’t use this woocommerce form field , because you can’t update price in checkout ,better than for you can add custom field in product page

/* Gift wrap price */ $additionalPrice = custom field value;

I will refer to product Custom field link