This quest was infinitely easier than the Kuzotz Region version primarily because the obligatory NM in Yhoator is a timed spawn and can be dispatched quickly without fear of links. Anyway, these quests aren't often done so if you're looking here chances are good that you're searching for information on where the items that will complete your quartet drop. I can't say for certain, but I believe that the items will only drop from the specified mobs in a given area and/or map. At least, that's how it seemed in the Quicksand Caves...
Rancor Handle: This guy. He'll most always be up unless he's killed out of boredom or curiosity by passerbys.
Rancor Mantle: Tonberry Cutter, mine dropped around H-8 on the second map (the area with coffers)
Rancor Tank: Tonberry Hexer, dropped at J-10 in Yhoator Jungle
Rancor Globe: Tonberry Harrier, dropped in H-10 of the first Ugglepih map. Much like this poor soul, I spent about 2 hours killing all the Harriers I could find in the temple (on both maps where they can be found) before getting the drop from the one Harrier in the Dragoon AF3 room. Ruling out coincidence, it's clear based on our two testimonials that he's the only Tonberry who drops this item :P
Once you've got the four items, make sure you've got a pickaxe in your inventory and head to E-8 on Lightningday and you'll find Peddlestox hanging out between the roots of a tree. Once you trade her the items and she marks up your map, head to the spot indicated, place yourself directly above it and... well... wait. For both beastman treasure quests that I've done, the ??? has been targetable only after a specific time. For this quest, I could target it at 23:30 (but not a second before) and didn't have to move from my spot to do so.
For all the frustration and hours of farming, I obtained: 7,000 gil, a Turquoise, a Bag of Tree Cuttings, a Silver Beastcoin, a Platinum Nugget and- thankfully- the Bitter Earring for a grand sum of ~212k. These are fun quests, and you can usually make a decent amount of gil if the "big", never-sells-on-AH item drops, but they definitely require a lot of patience and farming.