The school says the rule was introduced to respect students who are "sensitive to noise" but says the practice also "reduces fidgeting".
It is the latest in the series of school regulations to have faced criticism in the Australian media.
Hugs and Australia Day celebrations have faced bans at different schools around the country.
Earlier this week, an all-girls school rejected a newspaper report that said its teachers were asked to stop addressing students as girls, ladies and women in favour of gender-neutral language.
Elanora Heights Public School, on Sydney's northern beaches, announced the no-clapping policy in its latest newsletter.
"If you've been to a school assembly recently, you may have noticed our students doing silent cheers," it said.
"Instead of clapping, the students are free to punch the air, pull excited faces and wriggle about on the spot.
"The practice has been adopted to respect members of our school community who are sensitive to noise.
It said teachers would "prompt the audience to conduct a silent cheer if it is needed" and that they had found it to be "a great way to expend children's energy and reduce fidgeting".
A school in Sydney has banned clapping in favour of "silent cheering", "excited faces" and "punching the air".