Treat it like any other medical appointment. Inform the people that need to know (who that is depends on your company, but you should know) that you have a medical appointment and that it might take longer than expected or you might be out on sick leave afterwards.
Personally, where I live, vaccination is not a political issue. We do have peopl that think they get implanted mindcontrol chips, but no relevant political party would side with them. So where I live, it's perfectly fine to come forward in sprint planning and say "next Thursday I'll get my booster, I might be late and I might be out for a day or two after that if it hits just like the first two".
If you live in a country where this was made a political issue (cough US cough) you might want to be more guarded and just call it "a medical appointment". I would suggest you follow the lead of all the people that already got their first two shots. If they said it out loud back then and nothing bad happened to them in the company so far, I guess it should be okay to just say it. If they kept it secret or if anything unfair happened to them, just keep it for yourself and be vague.
In many countries (like mine) there are rules for the workplace. So no, nobody (except the designated health officer at the company) is allowed to know my vaccination status, but all workers that aren't vaccinated have to be tested to enter the premises. So... yes, you don't know my status, but you know if I'm being tested at the door. Which means colleagues know, they just don't offficially know. But again... where I live, vaccination status says something about vaccination status... and nothing else. You need to make your decision based on where you live.
As a general guideline whether I share private health information, I try to be helpful to my colleagues. If I'm on medical leave, I try to give them enough information to know whether they need to take action themselves. For example if I sat in a meeting with 5 coworkers yesterday and I'm out sick today, there is information that is helpful. Do I have something contagious? Did I hurt myself on company grounds, is there something dangerous they should avoid? Personally, if I had direct contact with a colleague and they are now out sick, I'm always relieved to know it's something where I don't have to worry about. For example, I have not to worry if they are getting vaccinated, or have a dentist appointment or they are seeing their orthopedic. That's nothing I can catch. Now if they slipped on the stairs or caught the flu, I might want to avoid that flight of stairs or disinfect their area. So if you want to be helpful, share information that is helpful or actionable. The fact that they don't need to take action can be good news, too.