"Placing an order" means ordering something. You can "place an order" for things like:

• takeout food
• stuff that you buy through the Internet
• a big shipment of supplies for your company

Grammatically, you place an order "with" the company that you're ordering from, and "for" the thing that you're buying:

I placed an order with Diamond Comics for 30 copies of the new X-Men comic.

However, when you're ordering food by phone, you can also say that the order is "for pickup", meaning that you'll go to the restaurant to get the food, or "for delivery", meaning that you'd like someone to bring the food to your home:

Hi. I'd like to place an order for pickup, please.