I'm not sure how you would define "legitimate" as, but let me try to be as comprehensive as possible with my answer.
Yes, a lot of retailers definitely use drop-shipping in order to run their businesses. It simply entails going into a contract with a supplier, and forwarding all your orders to them, so that they can fulfil these orders.
Few reasons why a retailer might use this method: 1. Decrease inventory- When the supplier is fulfilling all your orders, you do not have to hold any inventory, and yet you're legitimately running your business. 2. Quick Time-to-market - Dropshipping eliminates the time required for the product to come all the way from the supplier through the distributor to your inventory before you can sell the product. 3. Broader product offering - Again, due to no inventory, you can get a multitude of suppliers to dropship products to your consumers, hence also giving your customers a wider choice.
I'm the Head Strategist at Sellers Commerce, an E-Commerce Solution Provider, and hence I can tell you with experience that our retailers' first choice is always drop shipping. Sellers Commerce provides a way for suppliers to upload their catalogs in our cloud so that the retailers can simply subscribe to them, pull in the product data, and start selling. Now, this is where the drop shipping comes into play. Due to all communication being on the cloud, it becomes easy to enable drop shipping for these retailers, and hence as soon as they enable it, all orders they receive on the store get instantly sent to the supplier who then fulfils it.
I've seen many retailers shift from the drop shipping method to a more inventory-based fulfilment process when their operations got bigger.
So I would say that drop shipping is quite a legitimate business, just until you have the actual capacity to store inventory, if that makes any sense. Do let me know if you have any more questions about it, and I'd be glad to help you out.