First, it is "plate" not "place".
And I think "a rice plate" is not the same as "a plate of rice". The first you can order from a kitchenware shop, the latter from a restaurant.
"Rice plate" can mean "rice dish" / "plate of rice". It's a bit less standard but still correct. I haven't heard "rice plate" in the context of kitchenware.
In the case of the term 'rice plate' plate does not refer to an actual 'plate' but a type of meal equivalent to 'rice dish' which again does not specify the object the rice is served on but that it is a type of meal. 'plate of rice' specifies an acutal plate similar to a bowl of rice or a spoonfull of rice.
I am not sure how to express a 'rice dish/plate' in vietnamese. Maybe "ăn cơm cơm"? Probably not.
Rice plate can also be interpreted as Natt does, in which rice describes function. It is a plate for serving rice. But I think your use is the one I addressed above.
Note that gọi comes from Chinese 噲, Pe̍h-ōe-jī kōe (Mandarin kuài), but in Chinese it means ‘gulp down’.
Also, đĩa comes from Chinese 碟 dié.
So is "goi" the word we'd use to order or simply the description of the action? Is this what we'd say to a waiter when I order or what say when he has brought me something completely different due to how atrocious my pronunciation is :)
You can use gọi món (kêu món is commonly used in Southern Vietnamese) to refer to the action of ordering food although typically if you want to order you would use gọi món. "Cho tôi gọi món" is used to mean "I wish to order" even though the word for word translation is "permit/allow/let me (to) order food".
Do this group share one plate of rice or they order a plate for each person?
It's one plate/dish between them. If you want to say "each of us order(ed) a plate of rice" it's "chúng tôi mỗi người đều gọi một đĩa/dĩa cơm".