FrauSue�s answer is correct, but not quite complete. Her example of a catalogue is essentially dealing with the future: If you request a design, we will supply it. Since Apayala�s question is dealing with the present, only "by request" works, if, as FrauSue said, no requester is mentioned.
I will do that by request.
I will do that upon request.
("I will do that at someone�s request" doesn�t make sense, because the request hasn�t been made yet.)
The use of "upon" in a situation like this means about the same as "after", which is why it only works for future situations. "I will do that upon request" is the same as "I will do that after I receive a request".
I am doing that at his request.
I am doing that by request.
I did that at his request.
I did that by request.
I hope this is clear.
Bruce