You can't.
You need to equip a sword to get past Mido, you can't damage Ghoma without a sword, I'm fairly certain you can't damage the King Dodongo without a sword (it's impervious to bombs and I don't think the slingshot hurts him, but I'm not 100% sure about that one), etc.
You have to equip a shield, too, but you can just burn that up as soon as you find a Fire Keese or something.
For a three heart challenge, you might need to equip a shield to get to the Great Fairy on top of Death Mountain, especially on Master Quest. As a child, those rocks ALWAYS hit you. You can't avoid them, you have to block them. You need to visit that Great Fairy for the other Great Fairy to give you Din's Fire, which you need to open the door to the Shadow Temple. Though, I guess you can just skip that place and come back to it as an adult when you can dodge the rocks.
You also need to equip a shield to solve some puzzles (not to mention beat the boss) of the Spirit Temple.
So yeah. That challenge can't be done.
EDIT: Oh, pfft, if you're going to use glitches, screw it then. If anything, the game is easier if you start with the Megaton Hammer. (If that glitch even works in the DS version, which it probably doesn't.)