Well, if you pay yourself you fall under the rules of a “Privatpatient” (private patient, usually people who have private health insurance, not the social-security-related insurance).

Generally speaking that’s helpful because you’ll make the healthcare providers more money than a normal patient. I.e. you’ll have an easier time getting an appointment.

Also generally speaking the rule of getting a vaccine as a private patient is that you’ll have to do two things: buy the vaccine and then find someone to inject you with it. In case of Covid residents of Germany get the vaccine for free from the state (not from insurers, that is unusual), but I don’t think that would applies to you. Hence you’ll have to plan the logistics beforhand It’s possible to get the vaccine through a doctor who then vaccinates you and it’s also possible to get it from a pharmacist who then vaccinates you, but it’s quite likely that you’ll end up having to go to a doctor, get a prescription, go to a pharmacist, get the vaccine and then go back to the doctor to get vaccinated. I’ve done that and it doesn’t necessarily take more than an hour, but if you’re not in the country for long, you really want to plan beforehand. I.e. make sure the doctor has time and the pharmacist has the vaccine. How to plan does - just as @ebikefolder said - very much depend on the city you plan to visit. There’s no spoc for healthcare in Germany. You’ll have to find doctors and/or pharmacists via google (or specialized search engines like jameda.de).