I just got vaccinated this morning in the basement of the Plaza Lama located on Avenida 27 de Febrero. It is free and there was nobody waiting in line when I went. Maybe that was because of when I went: about midday on a Thursday. They are open till 3pm, and you need to bring your passport. You will receive both a vaccination certificate to show at the airport, as well as an appointment date for your second dose about 3 weeks later.
Below is an actual video of the plaza lama, just after I got my first shot: