Honestly I do more doging than blocking in Zelda games to begin with so no sheild runs aren't a problem for me and when I finish Skyward Sword (currently doing my first playthrough) I will do a no sheild run.
It's possible but hard. For most of the game you won't need it since regular enemys are easy but when you get to the electronic skelton on the Sand Ship it get very hard to do it without a shield. I tried Hero's Mode without a shield and I got to the sand ship boss and gave up. Everyone else is easy though.
The first time throw... I don't think I used the shield at all, I jumped back and side stepped the whole time.... Now after playing it a couple of times... I realize JUST HOW MUCH EASIER IT IS WITH THE SHIELD!!!
The enemies suddenly became laughable easy to beat... with right timing with the shield.
The only game I found actual use of the shield was the original LoZ. I all other games I only used it at certain parts of the game. I MM and OoT I took the hits like they were nothing. In WW I sidestepped and back flipped. In SS I have not really found use of it yet. In PH the shield didn't do crap. In FS it was almost always temporary. In ALttP I have not yet found much interest in using the shield. In TP I used it a little but not as much as I want to. So, as you can see, the shiels is almost useless to me. So to answer your question, yes. It is very possible to beat SS without the shield.
I can't beat Skyward Sword using a shield, i have too much of a tendency to move both of my arms while swinging thus raising my shield and stopping my attack and raising your shield against Demise is pretty pointless.