They can use the diner kitchen! There's no one in there. Where does the food come from? If someone orders food while you're in the kitchen, it just shows up in the little window between the kitchen and the rest of the diner when the little bell is rung, as if from nowhere. The waiter doesn't care that you're in there, and his dialogue will stay the same.

The kitchen is fully stocked with the basics, if outdated (that 1920s technology!). The stock will slowly replenish over the course of a couple days after you use something.