For example in medicine, you have requirements in many states for what constitutes a valid patient / client relationship and that this requires some in-person visits etc. does something exist like this in the legal profession?
I don't know of any regulations specific to the practice of law online.
A lawyer practicing online is going to practice under all the same restrictions as a lawyer practicing face-to-face. In the United States, that's typically going to mean some variation on the ABA's Model Rules of Professional Responsibility.
In particular, practicing online could raise issues of unauthorized practice of law, if you're somehow communicating with clients who are out of state about matters arising in a state in which you are not licensed to practice.
As for establishing a lawyer-client relationship, I don't know of any jurisdiction that imposes a face-to-face requirement. All that a lawyer needs to have a client is an agreement to provide legal services to that client; this agreement might be reached in a formal writing, orally, by phone, by text, or any other medium.
Providing legal services via video conference can be permissible for many kinds of legal practice, so long as it doesn't involve providing legal advise to clients where you aren't licensed to practice law, or aren't competent to provide legal advice.
This isn't true of all legal services, however. Outside, perhaps Alaska (which has specific constitutional provisions allowing for participation in government activities by video due to its geography), most civil litigation requires some in court participation (for evidentiary hearings), almost all criminal court appearances have to be conducted in person, and most consultations with incarcerated clients have to be conducted in person. Similarly, certain kind of due diligence in business deals, usually conducted by lawyers, requires an in person visit.
But, lawyers routinely confer with clients by mail, telephone, text message, email and videoconferencing and this is perfectly acceptable. If this can be done without risking loss of attorney-client privacy, it is usually fine.
The reason behind the expectation that physicians meet with patients in person is that the physician needs to physically observe the patient's body to evaluate them which until very recently wasn't possible with the necessary quality without an in person visit (and even a video conference is second best because it can't transmit smells which can sometimes be important for a diagnosis).
This concern isn't present in an attorney-client relationship in most cases, and no ethical rule prohibits having an attorney-client relationship without meeting in person outside certain kinds of litigation engagements.
No, you can practice law by phone or video conference.
I live in a small town and lawyers from different counties phone into the court when they can not be there in person. It has happened for years.