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.