The City Council approved moving the deadline back after the original plan would have had it take effect last week.
There remains pushback from the public and city employees over vaccine mandates.
Some expressed their concerns at Tuesday's City Council meeting.
"I believe in medical freedom," one speaker said. "It is our own right to decide what goes into our bodies."
City employees who don't get vaccinated by Dec. 18 face the possibility of losing their jobs. The council order does provide some exemptions for medical and religious reasons.
"It will be a condition of employment," Mayor Eric Garcetti said. "We want to keep every single person but to be very clear this is a condition of employment, to make sure that everybody - unless you have an exemption - is safe and interacting with the public."
Garcetti defends city vaccine mandate: 'I as mayor will never apologize for saving lives'
Also Tuesday, an FDA advisory panel endorsed the emergency use of Pfizer's COVID-19 vaccine for children ages 5 to 11. The full FDA still needs to give its approval, as does the CDC.
Seventeen members of the FDA panel voted in favor of the new recommendation, with one abstaining.
Doses given to children would be smaller than those for adults. Two shots would be given 21 days apart.
FDA vaccine advisers vote to recommend Pfizer's COVID-19 vaccine for children 5 to 11
Officials say so far they've seen more than 8,300 children hospitalized for COVID-19, and about a third of those had to go to the ICU. There have been more than 1.9 million in that age range who have been infected with the virus.
Pfizer says trial data shows one-third of an adult dose is nearly 91% effective against symptomatic illness in children. The FDA says the shots carry a rare risk of inflammation of the heart muscle and lining.
"The questions that I always ask are is it safe? Is it effective? And will it be impactful in the lives of children?" said Dr. Grace Lee with Stanford Children's Health.