How to acquire tan army key without purchasing ?

AFAIK, there isn't much way to earn free tan army keys. Best way to make use of the chest without shelling out money is by trading them in a special event like "Happy Purchase". Usually, you'll have to trade a number of item or resources plus 1 Gold to turn it into another useful item (e.g. 40000 Paint + 1 Gold = 1 Hour Build Speed-up.). We encountered an option of trading Tan Army Chest with Gold to a VIP item.

You may get a slim chance of getting a tan army key when you defeat "Victory" in Special Operation (the boss like spot on the 1st stage). Otherwise, random game events sometimes give them as rewards too. Hope this helps!