As far as I know, no quick way exists about farming these supplies. They're uncommon rarity drops.
Here are the supplies and how you get them:
-Heartwood (cutting wood)
-Bast/Clean Pelts (killing wildlife)
-Decorative Wax (found in crates, barrels, sacks and some other places)
-Mundane Rune (found from Runestones which are the rocks that glow)
-Alchemical Resin (found when you go to collect plants)
Now while no fast way exists for farming that I know of, the method that I found, used and liked was I would go to Auridon. On the Auridon island go all the way to the very bottom of the island to Vulkhel Guard. Now travel along the beach. You should pass Eastshore Islet (which you should also circle while you're there) and eventually end up passing Phaer. Now at this point you can either recall and start back at Vulkhel Guard or wait a little and turn around and go back the way you came.
This path is great because you'll run into all the things you need.
You can mine Iron Ore (which are the brown dirt piles or giant pieces of poop looking things) or Orichalcum Ore which is a two pointed green rock.
Beech ad Maple wood are very noticeable logs of wood along the shore.
You'll run in to bears and wolves that you can kill for pelts. (You can collect other pelts from them but only the two above count.)
Runestunes are everywhere. They glow red.
You'll come across barrels and crates which might have the wax.
Resin you get from plants and while they can be found, I didn't go out of my way looking for them. If I saw the collect prompt come up then I would check them.
I hope this helps. Grinding this should take maybe under ten hours depending on luck.
This is pretty boring as most grinding is, so I wouldn't try to do this all in one sitting or due to the random nature of obtaining the supplies wait till the last days of the challenge to start.