For vaccinations like the flu shot [1], I call them or send them an e-mail and then I can usually come by any morning I like without an appointment (they'll just administer the shot between appointments, since it does not take much time).

When we lived in Germany, we got TBE shots (area with a lot of ticks that carried it). IIRC we could also get an appointment in a few days.

(Though I think with vaccinations it's similar in the US.)

[1] Not eligible yet, so I pay 25 Euro to get one every year. Worth it for not getting the flu.