Campania is one of the Southern Regions, therefore everything to most foreigners is very backwards, there is no online booking system, there is no online health records and so on. so everyone carries there own health records with them when they go anywhere. Registering for health care is simple and they operate the voluntary scheme as well. They offer no information in English so you need to translate it or if you understand Italian then you will be fine.
Don’t misunderstand, health care in Italy is very good, but it’s very different depending on where you go. For example in Naples its substantially better than in Avellino both within the Region of Campania, one has online booking for things, one does not. Which is how most of Italy works, one town can have different health care than another if it comes under a different ASL (Regional Health Authority).
http://www.regione.campania.it/cittadini/it/tematiche/informazioni-di-servizio-5bk7hay0
Covid-19 Vaccine Booking System for Campania is https://adesionevaccinazioni.soresa.it/adesione/cittadino
English (UK) 
Italiano