Its analysis of Department for Education figures and local pupil forecasts found this could affect more than 125,000 children by 2022-23.
The LGA said councils needed powers to force academies and free schools to expand if additional places were needed and restoration of powers to build new schools where needed.
At present there is a mismatch between councils having a duty to ensure there are enough places in an area, but final decisions on new schools being made in Whitehall.
According to the LGA, a surge in demand has seen an additional 600,000 primary school places delivered since 2010, largely by councils using powers require their maintained primary schools to expand.
As this surge in numbers feeds through into secondary schools, more places will be needed but councils lack powers to deliver them.
The analysis showed that unless more secondary places are created, 12 local authorities will face a shortfall in 2018-19, rising to 66 in 2022-23.
Richard Watts, chair of the LGA's children and young people board, said: “Councils are working with one hand behind their backs to help as many pupils as possible receive a place at their first choice school.
"If we are to avoid this looming secondary school places crisis, councils need to be able to force existing academy schools to expand if voluntary agreement is impossible and must be given back powers to open new maintained schools themselves."
A Department for Education spokesperson said: “These figures are thoroughly misleading. Local authorities have a statutory duty to ensure that there is a school place available for every child.
“We have allocated £5.8bn of basic need funding between 2015 and 2020 to enable them to do this, and over 735,000 additional pupil places were created between 2010 and 2016. This money is given to councils based on their own estimates of the number of places they will need.
“We also know that a vast majority of parents are securing a place for their child at their first choice of secondary school - 83.5% this year, with more than 94% having received offers at one of their top three choices.”
