No.  Immunity from SARS-CoV-2 infection in children is not long lasting.  Immunity from SARS-CoV-2 infection appears to be related to how sick one gets, and children generally have less severe disease and therefore have less antibodies to protect them.   Antibodies made in response to the vaccine are much higher and block the virus so that it cannot infect people’s cells.  In studies at Seattle Children’s Hospital, antibodies after having COVID-19 go down pretty quickly over the first 6 months after infection. With this in mind, and knowing the safety data from the research, it is safer for children to get vaccinated rather than waiting to see if they will get COVID-19.
Among children who tested positive for SARS-CoV-2 infection and were then vaccinated, their antibody levels were higher after vaccination. Because of this, it is also recommended that children who have already had COVID-19 should be vaccinated as well.