My GP has an online tool to make appointments, and it’s great. I wouldn’t use it for genuinely private stuff because the company behind it is a bit shady, but booking an appointment to get vaccinated is generic enough for me to pick convenience over privacy. It’d be great if they could use open-source, privacy-respecting software, though.
You just need to add a thin layer plastic on the inside to prevent that from happening! Then it’s no longer plastic-free, but it’s still reusable and greatly reduces the amount of plastic needed compared to disposable cups.