Where/how does one get the flu vaccine?

I'm an exchange student who came to Japan at the beginning of this month for a year's study. When I'm at home in the UK I make a point of getting the flu vaccination every year, and I want to get it here as well, especially as I intend to travel in winter and will therefore be coming into contact with a lot of people in peak season. In the UK, I'm used to just walking into a pharmacy (drugstore), making an appointment and getting it a few days later, but obviously healthcare works really differently here, so I don't know where to get it and there doesn't seem to be any information from Google in English. I am registered with my university health centre, so is it there that I should go to ask about it? What is the procedure for enquiring? It's silly but I'm scared of walking into the wrong place and making a fool of myself. Any advice would be much appreciated.

I'd say your university health center would be the first place to enquire. Some companies offer flu vaccinations to all their employees, so there might be something similar at your university as well. Otherwise they'll be able to tell you who to contact. Likely a doctor.

Flu vaccination is very common practice in Japan, so you should not have a problem finding a place.

And regarding "looking stupid", well, you are here to learn. You are not expected to know everything about Japanese flu vaccination practice. So just ask. It's not a stupid question.

Enjoy your exchange year!

I agree that you should ask at your university, because more likely, universities have a list of doctors (and dentists) that its students can visit to get medical care for free.

Thanks very much for the advice and encouragement :) I went and got it today - the university health centre didn't actually offer it but they were very helpful and pointed me to a nearby clinic that did, and I went that afternoon and was able to get it immediately. No luck on getting it for free, but I wasn't really expecting to and I got it quickly and efficiently, so that's fine. One more thing checked off the to-do list!

there is no place where you can get it. it costs about 3500 yen.

where you can get it FREELY.

Cost variate from around 2500 up to 5000 yen.

Unless your company pays for it...

@likebike and were is mentioned that OP is working, so far only studying.
In general correct that the company pays or partly pays for the influenza shot.

I don't know about vaccinations, but I just thought that this would be a good opportunity for the OP to find out if she is one of the many students in Japan who can get free medical help. The following Japanese-language site is just for reference on this system.
https://kindaipicks.com/article/001160

I remember I never had to pay for my many visits to my dentist when I was a university student. The dentist happened to be so skilled, informative and friendly that I went and paid to the same one for decades after I graduated.