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 Early Lakebed.
There are several ways to enter Lakebed Temple early without the Zora Armor, mainly Pillar Clip.
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. If you've obtained the Master Sword early, it's possible to skip past the windy underwater portion of the first room by save warping.
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 strategies to battling Morpheel without Zora Armor, see the Morpheel-Page.
For strategies to battling Zant's underwater phasewithout Zora Armor, see the Zant-Page.