I found a video of a guy running through the game up to Ganon without a sword. Wikidot won't let me post a direct link, but it's at http://www.cinemassacre.com/2013/08/04/the-legend-of-zelda-nes-first-quest-swordless-run-with-mike-matei [Link courtesy of Gabe].
He states that you have to have the sword to beat Ganon but can go through the rest of the game without it— this isn't strictly true. It is actually possible to beat Ganon without the sword, but that's dependent on a glitch.
And if you don't want to spend three and a half hours watching someone play Zelda, the short version is that he gets some "secret to everybody" rupees first, then gets the blue candle, then gets some more rupees, then gets the damage-reducing ring, arrows, and bombs. In the first dungeon, he gets the bow and the boomerang, and uses the arrows to defeat the dragon. And then he does more stuff, because honestly that's as far as I got.
But it is a legit thing.