Although I wouldn't want you going behind your vet's back, no, you don't need approval to give it. Methylcobalamin, or Methyl B12, is available in many health food stores, and in some pharmacies, or it can be ordered online from several reputable sites. (Xobaline is one well-tested brand name in the sugarcat world. Zobaline is the new cat-specific version of Xobaline, made without fructose or any other sugar-based flavors.)
Important note: No matter which brand of Methyl B12 you buy, please make absolutely sure that it does not contain Xylitol (an artificial sweetener), which can be toxic to cats and dogs.