In many cases, you can start an affiliate business without worrying about the shipping. Often times, you can simply send prospective customers right to the merchant site and the merchant site will take and process the orders. In other cases, you process the order and then send it to a merchant who offers drop shipping. Basically, this means that you take the order and collect the money from the customer. Next, you send the order to the merchant or manufacturer, who packages it and ships it straight to the customer. You don't have to worry about inventory or shipping.
Make sure you understand all the shipping costs involved so that you can sell products at a high enough price to make a profit. Some drop-shippers add a service fee to each shipment. Be clear about the return policy as well, so you don't get stuck with items your customer has returned.