It is not currently possible to beat the game without Flippers.
This page serves to catalog the many different methods and techniques useful for completing dungeons and gathering collectibles without getting the Flippers.
Use the Crow south of the House of Gales to perform a damage boost off the ledge and onto the light blue raised section of land Link can walk on without drowning. From the top right portion of this area, use the Ice Rod diagonally to create an iceberg. Position Link just south of that iceberg and, as it it gets close to fully melting, use the Hookshot north to grab on to the dock's stairs.
If Link's positioning was correct, the melting iceberg will give Link some extra height (although not really noticeable) and allow him to get on the dock without falling into the water.
Fake Flippers can alternatively be used, though it is more difficult to execute.
There is a small swimming section in the basement of this dungeon that needs to be circumvented.
Either setup will work and the positioning does not need to be precise at all. Slowly inch Link to the right and he should fall down into the unopened corridor leading to the Master Ore. From there, you can simply walk south out the corridor (the wall will not stop you) and pull the switch to lower the water.
Then just back track to where the Thief Girl is being held to continue the dungeon as normal.
Perform Fake Flippers at either of the these locations:
Afterwards use Animation Storage for easy access to the dungeon.
This dungeon cannot be completed without Flippers
There is no known method of getting Link over the gap in the very first room.
It's still possible to do the Early Boss Key but it's no longer optional to use the Dash-Cancel Momentum to reach the ground on the right.
In the following room Link will have to merge along the walls from rafts to avoid falling in the water. Continue then as normal until the room with the Sequence Break. Link will either have to merge along the right wall and hookshot once to reach the stairs or traverse the hookshot targets while on the raft.
There is no known method of getting Link to the southeast door in the center basement room without Flippers.
Skipping the Second Miniboss Fight still works as normal. There is currently no known method of completing the room that follows.
It is no problem to hit the floor switch that opens the exit door, but in doing so Link becomes stranded and must either use a Scoot Fruit or death warp and retrace his steps to that point to continue. Reaching the first water-raising switch can be done by merging along the left wall and going all the way around the room. Presumably raising the water here would grant Link Fake Flippers. Using them, Link can continue as normal until the water is raised the last time. There is no way for Link to reach the door back to the main room from this point.
The last portion of the dungeon before the boss can be completed by just having Link merge along the right wall until he gets to the platform with the floor switch.
The Flippers Treasure Dungeon is a unique place in the game where you can use the rising water to activate Fake Flippers instead of screen transitions. To actually complete the dungeon and open the chest containing the Silver Rupee, you will need the "Nice" variant of either the Bombs or the Ice Rod.
Start by using thrown bombs to activate the two crystal switches on the right. Afterwards, activate the one on the left and the rising water will automatically give Link Fake Flippers (make sure Link is not standing on a raft when the water rises).
You have two options from here depending on which "Nice" item you brought:
Hitting the second switch will raise the water a second time and give you Fake Flippers again. Use Animation Storage one more time to get Link to jump up on the chest, which he can open so long as he's standing on the lower part of it.
Falling in the water will make Link drown and respawn where he was when the water level was raised a second time, allowing you to swim to the dungeon's exit.
A common misconception is that you cannot get the Pegasus Boots without first getting the Flippers. This is not true, you can get the Pegasus Boots any time after first bumping into the Shady Guy at Zora's Domain.
Only 92 Maiamai are currently obtainable, meaning that while you can upgrade all of your items to their Nice equivalents, you cannot get the Great Spin.
The Bottle acquired from the Bird Lover underneath the bridge is unobtainable, as entering the area underneath the bridge with Fake Flippers will lead to Link drowning/respawning indefinitely.
The Blue and Red Mail are both currently unobtainable as the Swamp Palace, and by extension Lorule Castle, cannot be entered without Flippers.
Every other Item or piece of Gear, including upgrades, is otherwise currently obtainable without Flippers.
A total of 92 Maiamai can be obtained without Flippers. This is enough to upgrade all inventory items, but not enough to get the Great Spin.
These 7 require Fake Flippers to collect:
These 8 Maiamai cannot be obtained without getting Flippers, as they're submerged in bodies of water where Fake Flippers cannot be activated:
The 85 Maiamai not explicitly mentioned here are all obtained as they would normally be.
The Heart Piece located in the cave under the waterfall near Zora's Domain is the only Heart Piece in the game that normally requires Flippers to access. Using either Fake Flippers or just Damage Boosting onto the nearby ledge will grant you access to it.
If you return to Zora's Domain to get the Flippers after Oren is kidnapped, a glitched cutscene will play instead where Oren is invisible and only one Zora Underling is present.
The Smooth Gem (or the Sold Out sign left when you purchase it) will remain at the Street Merchant's shop so long as you don't ever get the Flippers.