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.

The Â£28m project includes proposals for a major industrial and commercial development at Exxon Mobil's former site in Bowling.
A road would also be built to provide an alternative route in and out of West Dunbartonshire.
The outline business case for the project will be presented to West Dunbartonshire Council on Wednesday.
The proposals form part of the Glasgow City Region City Deal.
The plans would see West Dunbartonshire Council pay Â£3.8m towards the overall cost, with the rest funded by the UK and Scottish governments.
The project is expected to support up to 690 full-time jobs and generate about Â£19m each year for the West of Scotland economy.
If the outline business case is approved, it will then be presented to the council's City Deal Cabinet in April.
Infrastructure work could begin in early 2021 and be completed in early 2024.
Richard Cairns, West Dunbartonshire Council's strategic director for regeneration, environment and growth, said: "The presenting of the outline business case to council marks another big step forward for what is a key regeneration priority for West Dunbartonshire.
"The future for the Exxon site is an exciting one that we believe will bring about real change for the better in terms of jobs and the economy."
Moped gangs armed with machetes have been terrorising people in the streets close to Prince George's future primary school, locals have said.
[NEXT_CONCEPT]
Plans to redevelop a former oil terminal in West Dunbartonshire could take a step forward this week.