How many NHL players have (like Colorado's Nathan MacKinnon on 20th February 2015) have had only three shots in a game and scored a hat trick?
Hockey Reference was able to at least partially answer this; I only searched on goals=3 and goals=4 (so a 5+ goal game wouldn't show up).
However, it seems to be a very common occurrence. For example, it happened 7 times so far this season; Shawn Matthias (of Vancouver) scored 3 goals on 3 shots on 2/13 versus Boston.
According to my Hockey-Reference search, it occurred 233 times since 1987 (when their database begins keeping track). It also showed 10 4 goal-4 shot games. I didn't look beyond that.