There are more ways to defeat your enemies than with weapons. To get the Resourceful achievement you will have to do just that. This achievement will require some tedious strategy.
The Resourceful achievement requires you to complete a battle without using any weapon cards. Weapon cards are the most reliable way to deal damage to enemies but not the only way. You can use hazard cards, cheat cards, and even the enemy's own attacks.
There are two easy ways to get this achievement. You can do it near the beginning of the game when there are fewer enemies to fight. The downside of doing this is you will have fewer hazard and cheat cards to deal damage. As a result, you may have to rely on getting enemies to attack one another or plan your cards carefully.
Alternatively, you can wait until you have expanded your deck and filled it with hazard and cheat cards. This ensures that you can deal damage with each dice roll rather than having to wait for a few key cards to show up. You can still cause enemies to attack each other. And of course, using your hazards and cheats carefully will speed the process up.
For more information on Lost in Random, check out Should you give the Bloobs to Jared or Jored in Lost in Random and How to get the En Garde! Achievement in Lost in Random on Pro Game Guides.