See other threads regarding petshops. My only experience was in one about 7 years ago in Doha, and I have never set foot in one again. If you look at some recent threads you will find many people have had shocking experiences in some local petshops that sell live animals.

If its petfood or supplies you are after try Rider Trading, Carrefour, Landmark, etc. Most supermarkets stock pet food and some basic supplies. Rider tends to stock food and even more supplies and will often get particulars in if you have something in mind.

If its a pet you are after see threads of dogs/cats being given away. Or scan through this board looking for posts relating to Qatar Animal Welfare Society (QAWS.