I'm afraid you can't find any proper dog breeder in Hanoi.
You don't need any license or this kind of paper to adopt a pet. That's why no legal shop is opened.
We often buy pets from markets (which you don't like) and "buy" (not a trade, it's like a gift and you pay a little money to thank) from neighbours or relatives.
University of Agriculture has a department study about cattle. They may sell pets (including dogs) but I'm not sure.

I know of two puppies that need to be adopted. They were found abandoned in an alleyway.

PM me if you're interested.