Получить доступ к одному значению массива объектов php

Вам нужно зациклиться, чтобы получить все checkpoint_status

foreach($array['data']['items'] as $item){
   foreach($item['origin_info']['trackinfo'] as $trackinfo){
     echo $trackinfo['checkpoint_status'].PHP_EOL;
   }
}

Пример вывода: -https://3v4l.org/IM7I1

Вы можете использовать array_walk_recursive

 $r = [];
 array_walk_recursive($a, function($v, $k) use(&$r){
  ($k == 'checkpoint_status') ? ($r[]=$v) : '';
 });

https://3v4l.org/rCVEB

    $response = [
    'meta' => [
        'code' => 200,
        'type' => 'Success',
        'message' => 'Success',
    ],
    'data' =>  [
        'items' =>  [
            '0' =>  [ 
                'id' => 'b4a235596fd9550dfb69f181f4db007f', 
                'tracking_number' => '2649884668232181', 
                'carrier_code' => 'hermes', 
                'order_create_time' => null, 
                'destination_code' => null, 
                'status' => 'delivered', 
                'track_update' => null, 
                'original_country' => null, 
                'itemTimeLength' => 7, 
                'stayTimeLength' => 74, 
                'service_code' => null, 
                'packageStatus' => null, 
                'substatus' => null, 
                'last_mile_tracking_supported' => null,
                'origin_info' =>  [ 
                    'ItemReceived' => '2019-04-09 09:29',
                    'ItemDispatched' =>  null,
                    'DepartfromAirport' =>  null,
                    'ArrivalfromAbroad' =>  null,
                    'CustomsClearance' =>  null,
                    'DestinationArrived' =>  null,
                    'weblink' => 'https://www.hermesworld.com/en/',
                    'phone' => null,
                    'carrier_code' => 'hermes',
                    'trackinfo' =>  [ 
                        '0' =>  [ 
                                'Date' => '2019-04-15 11:51', 
                                'StatusDescription' => 'Posted through letterbox', 
                                'Details' =>null, 
                                'checkpoint_status' => 'delivered', 
                        ],
                        '1' =>  [
                            'Date' => '2019-04-15 09:45',
                            'StatusDescription' => 'Delivery will be attempted between 10:00 and 14:00 today',
                            'Details' => null,
                            'checkpoint_status' => 'transit',
                        ], 
                        '2' =>  [ 
                            'Date' => '2019-04-15 06:39',
                            'StatusDescription' => 'On its way to the courier',
                            'Details' =>null, 
                            'checkpoint_status' => 'transit',
                        ], 
                        '3' =>  [ 
                            'Date' => '2019-04-14 22:33',
                            'StatusDescription' => 'At the customers local depot',
                            'Details' => null,
                            'checkpoint_status' => 'transit',
                        ],
                    ]
                ],
                'destination_info' => null,
                'lastEvent' => 'Posted through letterbox,2019-04-15 11:51',
                'lastUpdateTime' => '2019-04-15 11:51',
            ]
        ]
    ]
];

Если вы работаете в режиме разработки, вы можете включить отображение ошибок на php и получить такой результат:

echo $response['checkpoint_status']; // won't work as response array has no checkpoint_status keyed array
echo $response[0]; // won't work as response array has no 0 keyed array
echo $response[1]; // won't work as response array has no 1 keyed array
print_r(array_column($response, 'checkpoint_status')); // won't work as response array has no column checkpoint_status

Есть много способов получить результат, один из которых приведен ниже:

$trackinfo = $response['data']['items'][0]['origin_info']['trackinfo'];
$checkpoint_status = array_column($trackinfo, 'checkpoint_status');
print_r($checkpoint_status);

Результат:

Array ( 
    [0] => delivered 
    [1] => transit 
    [2] => transit 
    [3] => transit
)