How would it be if Peter is the object of the sentence? If you are looking for Peter, does it also add the "t"? Does it work that way for names or brand names? If I am looking for a Pepsi would I say "Pepsit keres"?
Is it the natural word order? What about "Kit Péter keres"?
That one is wrong, unfortunately. With question words, the question word is usually immediately followed by the verb. So, you have two options here:
"Péter kit keres?"
"Kit keres Péter?"