"At request, XXX shall assist in the search of the cause of malfunction."
"At the Customer's request, XXX shall perform its services for a separate fee outside the standby time as well."
I'd like to know if "at" should be replaced with "upon." I don't think it should, but I'm not sure.
I would rewrite the second one as "At the customers request, XXX shall perform its services outside the standby time for a separate/supplementary/an extra fee".