My first thought was that only at is valid in OP's construction (which is a noun phrase, not a sentence).
But it's worth noting that in an actual sentence, it's just about possible to use a different preposition between look and how...
When as a therapist you see a client, instead of looking at how he or she fits the existing research, you should look for how the client disobeys the rules of research.
The reason that writer specifically uses for in the second instance is that it strongly implies focussed attention attempting to detect a specific thing. Looking at something can often mean little more than "allowing your gaze to rest on something, with no clear purpose or special concentration".
As @StoneyB comments, to look on is normally only used to mean either to stand by (idly) and watch something, or to have an opinion about something (to look on X as Y). Neither of which apply here.