After day 15, a moose will spawn somewhere on the map. In order to hunt it, you need to acquire a rifle. You can do this through the event "Spotted!" around day 12 by successfully chasing down the renegade, or by searching the train south of your camp (RNG will determine if an event triggers to find the rifle), or through Sir Edward's quest to find Betty. Use your workshop to fix the rifle and send out an expedition with the rifle and ammo equipped. Once you trigger the moose hunt event, select the first option to get closer and then select the option to use your rifle. You will most likely injure, but not kill the moose. So, follow the moose to the next tile and rinse and repeat until you have successfully hunted it.
See the video below for more information.