"Заказ" means an "order", like when you order some pizza or maybe jewellery for your shop.
Где мой заказ? = Where is my order?
Ваш заказ будет доставлен завтра = Your order will be delivered tomorrow.

"Delivery" as a rule means "доставка", like in the example above.
Мы ожидаем доставку товара завтра = We expect delivery of the goods tomorrow.
Мы не делаем доставку в ваш город = We don't deliver to your city.

There is a nuance with these words "доставка" and "поставка".
"Доставка" usually refers to logistic things, such as a truck, courier, route etc.
Мы сделаем доставку курьером = мы доставим это курьером = we deliver it by courier / delivery man
Цена включает в себя доставку = Доставка включена в цену = Delivery is included in the price.

"Поставка" means the one fact of supply. Very often staff recognizes goods by its supplies (by date, for instance). Also it's like a full process in general. "Поставка" is a bit abstract term.
У нас было три поставки этих товаров = We had three supplies of these goods.
Они поставят нам товар завтра = They'll supply the goods to us tomorrow.
Это самая долгая поставка в моей жизни = It's the longest supply in my life.

"Поступление" implies the moment when your goods were unloading/unloaded to your store.
Когда было поступление этих браслетов?
Эти браслеты поступили на прошлой неделе.