Because we can enter the Lanayru Province Early without the need to complete the the Goron Mines, the sequence of getting the Zora Armor is never completed and therefore the Armor is never obtained. Skipping the Zora Armor mainly consists of proper utilization of the Air Replenish Trick and Cutscene Diving.
By utilizing cutscene diving, Lakebed Temple can be reached without completing the Lanayru Twilight. The two main methods of doing this are either using Rupee Dive if you are playing on the GCN version, or Midna dive if you are playing on the Wii version. These tricks are used to dive out of bounds, where you can then swim to the Lakebed Temple loading zone, using the Iron Boots to regulate your position in the water. Alternatively, the temple can be reached by using the Air Replenish Trick to dive to the bottom of Lake Hylia after clearing the Twilight and blow up the rock blocking the temple.
The Pillar Clip method can be done without Zora Armor or the Air Replenish Trick by sinking to the bottom of Lake Hylia while it is under twilight (requires Early Master Sword). The clip through the pillar and reaching the loading zone before you run out of breath is extremely difficult.
There is now a set-up for pillar clip without zora armour created by dragonbane0 and is very consistent.
A rupee dive can only be done with a Blue Rupee or higher. The quickest way to spawn one is to dig up a patch of dirt close to the dive spot. However, A Blue Rupee will not always appear.
Once a blue rupee appears, take out the sword, hold L (sometimes helps get a better camera angle), and press B a split second before you get the rupee. The timing is hard. If you swing your sword too soon, you'll get the rupee in midair and the animation won't happen to where you can dive out of bounds. However, this is easier than standing on the very edge, not swinging your sword, and hoping the dive works. From there, swim your way into the tunnel.
Swim to the middle of Lake Hylia where the floating wooden platforms are located. Get on one of the platforms and use the Gale Boomerang to bring a wooden box to you. Get on the very edge of the platform, and prompt Midna. You drop the box, which will then push you off the wooden platform, causing you to cutscene dive out of bounds. From here, swim your way into the loading zone.
This is by far the simplest method, and can be done at any time when you have water bombs. Simply use the iron boots to get to the bottom of the temple and use the Air Replenish trick to keep from dying so you can blow up the rock blocking the temple.
After you have entered the temple, you will encounter that the first room is completely submerged, with the exit quite a ways out of the range of your air meter. Normally in a segmented Any%, you would simply save in this room, which would then savewarp you to the end of the tunnel. However, the first room can also be crossed using the Air Replenish Trick twice and the Rocket Link glitch to surface at the end. This method, though, is hardly ever used as you wouldn't yet have the bombs in the Any% segmented route.
Normally to get the Lakebed Temple Boss Key, you would need to navigate an underwater maze, which is, once again, out of the capabilities of your air meter. Though this room could probably be navigated by using the Air Replenish Trick, it would take a lot of water bombs and is unnecessary as you can skip the boss key.
Drop down into the center of the swiveling staircase room, and head toward a patch of vines. Once you're standing next to them, place a bomb behind you, and Vine clip through the wall. You can do this even after the water level is raised by using Water Bombs and Iron Boots. However, once you clip out of bounds, you must un-equip them by taking out another item over the iron boots through the item select screen. Once out of bounds, equip the Iron Boots to sink below the floor, then swim toward the center of the room. Once you reach the center, the game physics will start to behave erratically. If you pass underneath some open floor, the game will try to warp you back in-bounds. This is useful for getting high enough to reach the boss room trigger, and can even put back in-bounds inside the small room that's unlocked with the boss key. Once you're high enough, head to the center and drop down. If you don't have the Zora's Armor, it's highly advised that you equip the Iron Boots before dropping into the Boss's Room. If you drop into the Boss's Room and don't equip the boots before you surface, you will drown just before you reach the bottom.
For the first phase, Clawshot the eye, then do a quick-spin to finish off it's first form. On it's second phase, start off by setting up the Air Replenish Trick, and float upwards with the bomb for as long as you can before releasing it. If your within range of Morpheel's eye, equip the Iron Boots and Clawshot onto it. Hit it four times with your sword, then un-equip the iron boots by swapping a different item over them. This will cause Link to let go of Morpheel, and avoid being launched up in the water. Repeat for the second cycle, but this time talk to Midna when you take off your boots. Instead of sinking down again, target Morpheel and finish it off.
The only time after completing Lakebed Temple that you would normally need the Zora Armor is during the underwater battle with Zant. From the start of the battle, equip the iron boots and wait for the large Zant mask to emerge. Once it appears, use the Air Replenish Trick to refresh you air meter, and grapple Zant towards you. Once you have dealt him enough damage so he disappears. Once he has returned to the Zant mask, refresh your air meter once more. From here, swim to the center of the room and wait for the masks to arise, and refresh your air meter one last time. Finish him off and proceed with the battle. An optimal battle with Zant during his underwater phase would take 3 water bombs, but because of the randomness of the fight, it is highly possible that the battle will take more than 3 water bombs.