Immediately head to the secret room to the left of your starting location. Teleport two of the teleport bombs and the master key, but not yourself. Leave the gems alone. Then kill yourself. These three blocks will still be in the stack, yet you can retrieve more of them right away. By repeating this process several times, you can get quite a few teleport bombs. You don't actually need more than three, but you may want to keep a few more. Teleport bombs are going to be very hard to get later on. If you want to play things safe, you can also use this opportunity to get more master keys. About two or three should definitely be enough, though. Also, make sure to teleport yourself so that you don't have to unlock the O door.
Next, put a blue cube and the yellow bird in the animal collector to get the A key and unlock its corresponding door. You can enter the village main screen as often as you wish. Just make sure to only leave it by teleporting or dying. Also remember not to open any of the doors here unless told. This means no shopping whatsoever. Do however enter the H tower to retrieve both the H key and one of the inventory books. Now put four random blocks in your inventory. Climb on the library's roof, then jump and drop all four blocks from your inventory to gain enough height to reach the top of the P tower and get its key. For the L key, climb on the credits building's roof, then throw an active teleport bomb towards the key just before denotation. The walls around it won't shield it from the explosion.
Now you can simply retrieve the animals as usual. Access to the H, P and L areas is all you need for that. It's highly recommended that you visit the illegal gun shop in the L area. To get there, take a master key and $40 with you, carry a blue or red bird to the big wall on the left, jump and have it flap as much as it can, then quickly put it into your inventory and out of it again, then have it flap some more. Repeat this process until you've reached the door. It's locked and requires an O key. Being the only locked door not connected to the save room, this challenge's restrictions don't apply to it. So use the master key and pay $40. This gives you a three item box with a teleport gun, use gun and violet gun. With exception of the violet gun, you'll need all of these. Also, don't lose the three item box you get after collecting enough animals. Otherwise, you'll have to buy another one from the illegal gun shop (which requires another master key) or find one somewhere else. Furthermore, you'll need a teleport box. You'll find one in the room in the H area to the right of the skull block. This room is cursed, so use a teleport bomb.
After getting the zoo ending, it's time to make some final preparations before going for the vault ending. In addition to the aforementioned teleport gun, use gun, teleport box and the two three item boxes, you'll also need a teleport bomb, a blue, green, white and black gem, and a blue or red bird. With exception of the bird, you need to cram all of this into your inventory using the three item boxes. Then use the bird to fly all the way up to the trophy room's back entrance to the left of the village main screen. You don't need it anymore after that, so leave it behind and go through the door. With this, you've used up the second time you're allowed to open a door on the village main screen. It's all over if you screw up after this.
Now, drop the gold trophy and build a tower with two gems right below it. Then throw the teleport bomb onto the tower, shoot it with the use gun and make sure you're not hit by the explosion. Next, drop the silver trophy and build a tower with some gems slightly to the left of the valuable collector. Once it's high enough, jump from it, then throw the teleport gun against the ceiling to make it shoot the silver trophy. Finally, use the teleport box to bring back everything you teleported, then put all the valuables into the collector before entering it yourself.