Please remember that I am presenting this from the US, so there are other countries where this information is not correct. Always do your research!
The easy answer is "NO", there is no requirement (in the US) that a service dog team have a note from their licensed medical professional to have their service dog. Likewise, there is no "certification" that certifies the team as a legal service dog team. So, all those ads you see online are complete crap, even the ADA (American Disability Act) specifically points those groups out.
"There are individuals and organizations that sell service animal certification or registration documents online. These documents do not convey any rights under the ADA and the Department of Justice does not recognize them as proof that the dog is a service animal."
With being said let's look at some of the nuances of having a letter from your licensed medical professional, and why it IS important to have one.
Housing: If you are renting you and your dog, probably, must conform to the FHA regulations for "assistance animals" which typically require "Reasonable Accommodation" paperwork and specific paperwork from your licensed medical professional. I have a more in-depth article about this: Service Dogs and Apartments (US).
Flying (domestically): To fly domestically, here in the US, with your service dog there is a form that must be filled out "DOT Service Animal Air Transportation Form and a Service Animal Relief Attestation Form". This requires trainer information, vaccination information, and you guessed it, a note from your licensed medical professional.
Work: Many of us take our service dogs to work. There are many steps that a typically made to do this, and every company is different. The base requirements are typically the "Reasonable Accommodation" paperwork and specific paperwork from your licensed medical professional.
Court: While many people tend to roll their eyes when I bring up this reason, it is something that must be thought about. Every one of us is only one small public accommodation conflict from having to go to court to prove our legitimacy to the court. All it takes is a simple summons from a law enforcement officer. Things like your doctor's note, training logs, letters from trainers and friends about your dog, and certifications from things like the CGC line all make something like this much easier.
My suggestion for service dog teams is to have a file with everything you can think of regarding you and your service dogs. While you may never need it, it is much easier to have everything in one place so if that time comes when it may be required it is quick and easy to retrieve.
