Frequent vomiting in uni shared bathroom - will anyone notice/care?

The way my uni does it you basically live in halls the entire time you're there and you ballot for a room so there's a chance you might not get your first pick. I picked a room with an en suite but there aren't that many available and obviously people with medical needs get them preferentially so I might end up having to share a bathroom.

Would the people on my floor be likely to notice if I was throwing up more days than not in the shared bathroom. It would be at night time, usually around 1-2am? And if they did notice, would they be likely to do anything about it like reporting me to uni? Or would they likely just mind their own business? I'm a major introvert so it's very unlikely I'll become close enough with anyone that they'd feel comfortable asking me awkward questions. I don't care if people gossip about it, just as long as they don't bring it up to my face.

It's entirely possible. You can't predict the behaviour of other people and I wouldn't base your accommodation decisions on what flatmates might do. I suspect some will notice, plenty of students are still awake at 1-2am, either working on assignments or just coming in from a night out. So they might hear or smell what's going on. But that doesn't always equate to action and that very much depends on the personalities of the people you may share with - which you can't know about at the moment.
And since we can't predict the behaviour of others, we can't guarantee someone wouldn't be willing to talk to you (about this or anything else).

Have you considered asking for your medical situation to be taken into account in room allocation?

I don't think getting so drunk I throw up is a medical situation lol

If you are getting so drunk you throw up "More days than not" that is very definitely a medical situation - from whatever the reason is behind you doing it, to the physical consequences of that much alcohol and that much vomiting. It's not 'lol' - and I think you are a bit worried about it yourself or you wouldn't have asked this question (not unusual for people to be a bit tangential)