This is a dialogue from Death Note. A guy is being asked to analyze some items.
Guy 1: これの鑑識お願いできますか?
Guy 2: 鑑識には顔が利く、徹底的に調べさせます
I don't understand what the causative is supposed to function as here. Nobody is being made to do something, nor allowed to do something, Guy 1 simply asks Guy 2 to analyze the items, and Guy 2 accepts and says that he will do it thoroughly.
My guesses are that, perhaps the usage here is similar to させてもらう, as in (I will take the liberty of) kind of meaning, and since Guy 2 is in a lower position here, it sounds even more appropriate (I will take the liberty of letting myself do the investigation).