No, sorry. You should not buy a puppy from a therapist. It would be unethical, IMO, for a T to sell any kind of side-business product to their clients. You should not be buying ANY product from your therapist other than the one you're paying her for —* help in processing your own problems and issues.

It seems, as others have said, that this dog is a way to acquire a different kind of relationship with your T. That IS understandable most anyone here would love to have some kind of outside relationship with out Ts, but you can find a dog elsewhere.

You might bring up the topic and see if she had input on getting a dog and why you would like one.
You might also be prepared to talk about why you Googled her so extensively, i.e., your understandable and very human need for more connection and be prepared to work through that.

Or try a pure breed rescue group if you know the breed you really like. I do not like certain breeds and their traits - so I have one adopted from a rescue group who does the type I want and I have one from a breeder. I have also been a foster home for the type I like. Also - I sent the breeder some pictures as the puppy grew up - but I got one as a pet not a show dog - the breeder, while polite and a nice guy, does not try to keep in touch with me about my (beloved) dog.
Being a foster dog care giver might give you a start and you can see if you really want a dog that way too. And research the breed or breeds you might be interested in - they are not all alike. For example, lot of people love terriers - I could not live with one at all. The same is true for me about labs and golden retrievers - great dogs for some one who is not me.

I think often the boundary thing gets a little over-played around here and I really don't think is that big of a deal.
But making sure you get the right pet that you will take care of for its whole life is super important - I think.