Yes, I finished the first quest of Zelda without a sword years before I even had internet access. I've attempted the second quest, but was never able to finish. In theory, you play through most of the second quest without a sword anyway, due to those goddamn red bubbles.
Technically, the only way to BEAT the game without the sword is to use the potion glitch, since the sword is the only thing that can harm Ganon