I'm a totally blind software developer. I used two sites to sign up for a vaccine.
The first site was a covid vaccine specific website. I was unable to use it with my screen reader. I can't remember the exact issue but it was something to do with a popup my screen reader could not interact with to select a date. Luckily I had family who were able to fill out the form for me with in a couple hours of opening availability. I was fortunate that my family were not in frontline jobs and could be at a computer when I needed them to be.
The second site was a mobile health clinic that was targeting disabled and at risk individuals. That worked fine because there goal was to get vaccines in arms and not worry about the technology. They used signup genius which had no accessibility issues for me.
I understand that accessibility is not always easy. I don't expect Xbox Live cloud to somehow magically make first person shooter games accessible. There's no reason in 2021 that a site that does nothing but have you fill out a basic form with the ability to select a date and time should be unusable with screen readers.