Yeah, I figured nobody was going to try this. Oh well, here's the solution. It wouldn't surprise me if this challenge was actually something Nifflas had envisioned, given all the convenient things he added to certain parts of the game for Uurnlimited that make it possible.
Go through the O area until you reach the part with the aliens. There's a hidden door above them that you can reach by using an alien to shoot most of the cubes blocking it. It's the easiest when jumping from the middle alien with just the right timing.
Once you go through the door, you'll find yourself in the Z tower. There's no need to actually go through it, but push the teleport box there onto the teleporter. This'll cause two power bombs to be teleported as well. They're pointless, but you can discard them later. Now go through the door next to the one you came from. You're now above the village right next to the JTC area's back entrance. Go there to get 17 coins, then return and fall down to the village.
Get three more coins in the village, pay a quick visit to the L tower just to teleport the inventory book, then buy a use gun and shoot the gems in the gem shop. Next, head to the box shop and shoot the teleport box to retrieve the inventory book from the stack and read it. Then shoot the teleport box again to retrieve the cheaper teleport box and put it in your inventory. Finally, buy a purple gun and reset. (You can also just buy the expensive teleport box instead, but then you'd probably have to do an extra reset just to get the purple gun.)
After each death, you have to go through half of the O area and reach the hidden door again, but this is now easier thanks to the purple gun. Return to the village and use the use gun for another gem shop robbery, then buy the blue orb. Head to the seemingly empty room in the middle of the connecting passage between both ends of the village. Use the orb on its right side to reveal a hidden door.
The area it leads to is rather dangerous. You can theoretically make it through it without bringing any items, but I'd strongly discourage that. The purple gun helps a lot here. Once you make it to the end, you'll find a bunch of goodies, but also a teleport drone. Destroy it immediately. If it shoots you, you'll end up in the save room and fail the challenge. You might misaim and destroy some of the loot, but that's a lesser evil compared to having to start all over. As for the loot itself, there are gems of all the colors you need for the vault ending, so if possible, teleport one of each color. If not, there are plenty of other blue and green gems you can find in the village, while the white and black ones can be purchased or stolen via teleportation. Black ones can also be found in a hidden room at the very left of the O area, but getting there takes some time. What really matters is the reusable three item box. It has a warp block, teleport gun and teleport box in this order. You already have a teleport box, and a teleport gun can be purchased or stolen, but the warp block is essential for this challenge. Now reset.
Head to the JTC area, drop the gold trophy, then squeeze the warp block through the gap to the right of said trophy's room. Shoot it with the use gun to be warped there. Drop a random block to get up to the gold trophy and teleport it or put in in your inventory. Put whatever you dropped back into your inventory unless you no longer need it, then reset.
If your stack is still missing some of the four gems you need, teleport them and reset again.
Finally, return to the JTC area, drop the silver trophy and teleport it by throwing the teleport gun against the ceiling. Now put all the gems lying around and the bronze trophy in the collector along with the missing items from your stack and inventory along with yourself.