The CDC does not authorize shots or vaccine distribution. The FDA authorizes vaccine distribution, or distribution of medicines in general. A medicine is authorized for use in a certain way, which is an "instruction" directed at the medical professional. So-called "off-label" use is discussed by the FDA here. Such use if legal, it simply stretches the limits of what the FDA was allowed to approve (e.g. was found to be safe and effective for X, was not similarly tested for Y). Patients are free to take whatever medicines they want if they can find them, and if they are not on the Controlled Substances list. Hence it's legal to take DMSO, but it's not legal to peddle it as a cancer cure.
The prescription regime for covid vaccinations seems to be somewhat relaxed, in that it is likely that many people get a vaccine without a (meaningful) doctor's exam, instead, you show up and get the shot. This is normal with flu shots. The upcoming 3rd shot approval is addressed to doctors, so they can then recommend third shots.