I have already played Skyward Sword several times, I just can't stop playing it. Each time I've played through it I have tried something different. It is indeed possible to beat Skyward Sword without a Shield and Hero Mode too. It does make the game much harder, but if you're up for the challenge then I suggest you try beating the game shieldless.
During my first playthrough of the game I didn't ever use a shield. I got used to dodging the enemy attacks instead of protecting myself with a shield, and it worked very well indeed. However, in my second playthrough I started using shields and I realized how much easier the game got.
So yes, I'd definitely say it is possible to play though Skyward Sword without a shield, you just need to get used to it.