You will need:
-lots of wet snow
Note that this requires very wet snow, especially for the cannon.
The wetter the snow is, the easier it will be to get the protuding mouth of the cannon to stick.
If your snow isn't wet enough, you can just support the entire barrel on a block of snow.
Also I wasn't planning for this to be an instructable so I didn't take pictures of every step.
I just made some 3D models in Blender. Because my computer was having issues, they didn't turn out exactly as planned, but they should still give a general idea of what's going on.
Step 1: Create the Cannon Base
Turn the sphere into a rectangular prism.
The dimensions should be about 1 foot by 2 feet by 1 foot.
Step 2: Form the Cannon Barrel
Roll another snowball about 1 foot in diameter.
Set them next to each other and pack snow in between them so
they form a tube with rounded edges.
Set the tube on top of the block. The tube should be about flush with the sides.
Now comes the more difficult part.
Start packing snow to the front of the barrel until it protrudes about 6 - 8 inches
in front of the block.
Then hollow out the middle of the cannon.
Keep the hole small. about 4 - 6 in.
Don't dig in too deep, max. 1 foot.
Now, the most difficult part.
Take small handfulls of snow and pack it onto the front of the barrel.
Go all the way around, then start another layer.
Do another couple layers, but remember that the more layers you do,
the sooner it'll collapse.
Congrats, you're done the hardest part.
Now just pack on snow where it's needed, and smooth it out.
You can also shave down the sides of the block for a more realistic appearance.
Step 3: Add Details to the Cannon
Add a small sphere about 2 inches in diameter.
To add a fuse, just poke a hole in the top and stick a small branch into it.
You can also add a small pile of cannonballs, like I did.
Step 4: Build the Snowman Victim
The second snowball has to have a hole that lines up with the cannon mouth.
The first snowball should be wide, but only about a foot high.
The second snowball should be higher and narrower.
Prop the second snowball up on the first, and pack snow around it
to secure it in place.
Carefully tunnel through the center of the middle snowball.
Don't make the hole any larger than 6 inches, as it could cave in when you put the head on.
Now make the head. Try to make the head to look as if it's looking down.
Don't make it too big, or it'll cave in the cannon hole.
Step 5: Add the Finishing Touches
I used hockey tape for the eyes and mouth.
I just pressed the tape into position and packed a tiny bit of snow around the edges.
For the nose I used a marker because there weren't any carrots in the house.
The tricky part is to get the snowman to look like he's staring at the hole in his chest.
For the arms I just used straight sticks I found lying around my yard.
Don't forget to add the cannonball lying in front of him.
Some details I didn't add:
-Red food coloring around the cannon wound
-Wheels for the cannon
-Carve away the back top portion of the block to make it look more like a cannon mount.