Is there any substitute for a grater?!
Use the vegetable peeler, make thin long slices lengthwise, then with a sharp knife start cutting until you have a very fine chop, it can be done with some patience.
Vegetable peeler will work best. Once you peel all of them just start chopping the pile until they are the size of grated carrots. Your other option of the blender may work too, but I foresee that going more towards carrot juice and chunks and not necessarily shredded or grated.
i would call a neighbor and see if you can't borrow their grater.
otherwise, you could probably do use a knife and do a fine julienne of the carrots and then cut them into as small a piece as you can.
I have a Pampered Chef food chopper. Do you have something like that?
if the veggie peeler doesn't work, try putting them in the blender. they be "grated" but they will be chopped up finely. if you are just using it in a cake it should be fine.
Do not blend. That will give you carrot mush.
You can peel then chop as another sposter suggested but I recomend running to the store and picking upa cheap grater.
you could chunk up the carrots
add water and carrots to the blender after they are small enough pour off the water pat dry and use