Dogs and cats have an essential amino acid profile which means no vegan/vegetarian diet can meet this requirement.
There are vegan/vegetarian diets available on the market which have been put together by parties who believe in such, but none are veterinary approved and none should be fed to your pet.
The only way that your pet can get the amino acids they require for life is from what they eat and it needs to be a balanced diet which contains some meat.
Have a look at these articles for more information:
http://www.nzherald.co.nz/lifestyle/news/article.cfm?c_id=6&objectid=11561354