At my house we have a snowball maker which you put ice into it and shave the ice to make a snowball, after that you pour the flavor onto the shaved ice. I was wondering if I’d be allowed to shave the ice, put the flavor on and eat it all on Shabbat.

One of the 39 labors forbidden by the Torah on Shabbat is "grinding". This is defined as grating orslicing into small or thin pieces. The most severe and inclusive injunction is against grinding uncooked vegetable produce. However, use of a utensil specific to grinding - like a cheese-grater or a mill of any kind- is forbidden even if what is being ground is not agricultural produce(Shulchan Aruch, Orach Chayyim 321,10; Mishna Berura 36). Shaving the ice on Shabbat in a utensil designed for that purpose (as opposed to using a regular kitchen knife) is therefore forbidden. If you can't prepare the ice before Shabbat, why not enjoy the snow on Motzei Shabbat for Melave Malka?