The remaining four clinics, including the Orchard Rd Vaccination Hub, are not offering bookings until next Wednesday and Thursday.
CDHB Senior Responsible Officer for the COVID-19 response Dr Helen Skinner said while not mandatory, they strongly recommend that parents make bookings for their children's vaccination.
There were 1,816 slots available for children in the wider Canterbury area on Book My Vaccine as of Monday afternoon up until the end of the month, she said.
"These are however being booked very quickly. 2,671 paediatric bookings were made yesterday in Canterbury.
"We are supporting clinics to increase capacity and we ask parents to keep checking the Book My Vaccine website as new appointments are being added," she said.
A caller on Canterbury Mornings with John MacDonald on NewstalkZB this morning said she had been driving around with her two children under 12 in Christchurch but could not find a clinic to vaccinate them.
"We went online and several places are listed as being available but they don't have their supply yet. We walked into two and they didn't. We are just feeling quite frustrated," she said.
More than half a million doses of the vaccine arrived in the country earlier this month, ahead of the roll-out covering some 476,000 eligible children across New Zealand.
While there are no plans for a school-based immunisation programme, schools are being considered as community vaccination sites.
A parent, caregiver or legal guardian needs to accompany the child to their immunisation appointment and provide consent for them to be vaccinated.
As with adults, children need two doses of the vaccine to be fully protected, and the ministry recommended these be given at least eight weeks apart - although the interval could be safely shortened to a minimum of 21 days if needed.
