At is the better word to use here. At carries a lot meanings related to being near its object. In this context, at means physically nearby, but also carries the connotation of (in the words of the OED) "some practical connection with a place". So not only does this use convey that the left turn is to occur physically close to the stop sign, it implies that the stop sign has a connection to the action of turning.
As the listener is assumed to know, the stop sign would be placed at an intersection, and the understanding is that the person following the directions would realize that they are supposed to turn left onto the road crossing the intersection. In contrast, we wouldn't say near the stop sign, even though it is technically true, because it doesn't have the implication that the stop sign is an inherent part of using the road.
As far as on goes, I think a case could be made for it by using the sense in which on refers to a point in time, or a point in a series of events. It would mean something like "on encountering the stop sign". The problem with this is that on has the very commonly used meaning of being physically in contact with, or on top of its object. (Obviously, being physically on a stop sign is far enough away from a likely instruction that it wouldn't cause anyone real confusion.)
For all I know, "on the stop sign" might be idiomatic in some regions, but I think "at the stop sign" is what is generally used, as well as being more correct. By the way, at is generally used in the same way for other indicators in directions: "at the curve in the road", "turn at the mailbox", "at the fork in the road", "at the third house on the left", etc.