Residents claim they have seen gangs wielding weapons speed through pedestrianised areas near Thomas's Battersea School in south-west London.
CCTV footage shows them weaving among restaurant tables, threatening diners and passers-by.
Prince George is due to start the Â£6,110 a year school in September.
"There are break-ins every night, residents are being terrorised every other day either by riding on pavements or at people," said one local, who was too scared to be identified.
"When police are called their response is 'there's nothing we can do'."
"The police don't chase them so it's almost a licence to commit crime," said another resident, who also wanted to remain anonymous.
Staff at Thomas's declined to comment on the moped incidents or on any security measures being put in place.
CCTV footage gathered by the Press Association from business website LondonLovesBusiness.com shows one incident where thieves on a motorbike tried to kick someone off their bicycle before breaking in to a car directly outside the school on Battersea High Street.
Residents said the gangs, who are believed to live locally, hide their faces with balaclavas and are dressed in black. The vehicles are believed to be stolen and the number plates are either blacked out or removed.
Insp Guy Osborne, of the Battersea Safer Neighbourhoods Team, said Wandsworth Police were "very aware" of moped-enabled crime, and called the issue "a borough priority".
He said officers were giving crime prevention advice and were working with moped manufacturers to make them harder to steal.
He added that incidents involving weapons had not been reported.
Moped-crime is a London-wide issue which is increasing.
A BBC investigation found that crime involving mopeds in the city had increased by more than 600% between 2014-16.
On Friday, moped gangs armed with hammers were photographed outside the BBC's Broadcasting House in central London and on Sunday, the Met released footage of a gang knocking down and breaking the leg of a tourist as they tried to steal his watch.
Moped gangs armed with machetes have been terrorising people in the streets close to Prince George's future primary school, locals have said.