"Who is Péter looking for?"

Translation: Péter kit keres?

[Thomasg88]

Is it the natural word order? What about "Kit Péter keres"?

[vvsey]

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?"

[SantiHM23]

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"?

[ssevi]

Yes, "Pepsit keres" is ok.

And if Péter were the object, it would be:

Ki keresi Pétert?

[ssevi]

Maybe I should have written was instead of were... hence the condition can be real. :)