Because I keep hearing people say that they can't book any local appointments on the website...
Nearly all vaccine hubs are now doing walkups. You/your young adult kids don't need to book. Just put your postcode into this govt website, and it will tell you where you can get jabbed straight away, without an appointment. Even if the same hub doesn't appear to have any appointments on the main booking site. If you/they have an NHS number to hand, they'll appreciate it when you get to the hub, but if you don't, they'll still be able to find you on their systems (I'm a volunteer and when I'm on reception I book walk-ins as well as appointments, with or without the NHS no)
www.nhs.uk/conditions/coronavirus-covid-19/coronavirus-vaccination/find-a-walk-in-coronavirus-covid-19-vaccination-site/