I don't think so. As far as I am concerned you would need the sword to be able to beat the first dungeon and all the boomerang does is stun and you need to kill everything to get a key, or open a secret passage. It isn't until a couple dungeons later you get the bow but i am sure you would run out of rupees real fast using just arrows alone.
In the 1st quest it is impossible, for a Gleeok blocks your path. The only way to beat it without a sword is to get the Magic Rod, which is in Level 6. But another miniboss Gleeok blocks it, so you can only play up to Level 4 without a sword in the first quest.
However, it is possible to beat the entire 2nd quest (except for Ganon) without a sword by using bombs, Magic Rod, and the candle. But this requires a lot of sequence breaking (such as going to Level 6 and 8 early)
There are secret spots that have rupees that you can buy the blue candle and bombs. Then there are other weapons you could use such as the bow, magic wand, etc. However it is impossible to beat Gannon, but it is possible to get to Gannon.