At our (Montessori) parent toddler group I've been privileged to witness children much younger than Otis cut, slice, spread and grate. Capable, young (I think the youngest is 18 months) children. With parents willing to take inspiration and guidance from our teacher.
Otis has never hurt himself grating or peeling. But it doesn't happen in any haphazard way.
Clearly demonstrated. Closely supervised. Often working side by side.
We've tried a lot of peelers over the years and I've always found the horizontal peelers work best for children.
Otis first started grating and peeling carrots (for Caspar's school lunch) however I believe the easiest way to start grating is with cheese or even soft soap. With peeling it's easiest with cucumber or zucchini, where the peel is thicker (than carrot for example) and clearly visible. And it's even better if the grating and peeling has purpose. Grating soap for making soap balls or hand washing. Grating carrot for sandwiches. Or peeling cucumber for snack!