According to the government website you can just show up at the vaccination centre, explain your child's situation and they will be able to make a booking on the spot as a walk-in. It worked for my kids.
This works for other situations too where you can't make a booking. For example, my wife has an HKID but because her shots were all given overseas she couldn't make a booking on the website. She was able to do so as a walk-in after explaining her case.