This looks right! Another way to think about this is the first friend orders and has $12$ choices. The second friend can then order anything that the first friend didn't, so they have $11$ choices. The third friend similarly has $10$ choices and the final friend has $9$ choices.
So by the multiplication principle, the total number of orders is $12 \ast 11 \ast 10 \ast 9$ as you stated.
As someone mentioned in a comment, it is worthwhile to note this is different from asking the question: "What are all the possible ways to choose $4$ distinct dishes from the $12$ possible dishes?" Since in your situation, we do care which friend gets which meal.