So how many vikings do you have? 10? That means you can take on a hundred or so zombies, right? What happens after that?
Your axe may not run out of ammo, but it gets a lot more tiring swinging one around than firing a gun, and if you run out of ammo, you made a mistake somewhere. Probably when you didn't deck yourself out with as much ammo as possible. Lets say you have an M4gery, the gun weighs 6.4 pounds loaded, each magazine weighs about a pound, you could probably carry around ten of them on a vest, if not more. That is a lot of shots, and if you run into zombies, your plan should be to fight your way back to safety, which is going to include even more ammo.
So, you are carrying about seventeen pounds plus the weight of the vest, probably just a pound or so, and the weight of whatever armor you are using, probably a thick coat and pants, one of those riot helmets would be awesome, but not likely to have on hand. So, less than 25 pounds, which is around what a highschool student carries around all day, and for students, the weight isn't distributed as well. How much does your gear weigh, and how long can you fight before exhaustion kicks in? Or before you slip up?
And you seem to be thinking that you can only be able to fight at close range or long range, you ever been hit upside the head by a rifle butt? Me neither, but I don't want to volunteer for the experience.