Residents have said that, over the past few months, the issue has become 'commonplace at weekends' and 'poses a risk to the safety' of people visiting Kingley Vale.
It is alleged that the vehicles are accessing the site from Stoughton on the farm track that passes a war memorial.
One resident, whose Labrador was nearly run over and witnessed small children being endangered, said there are no gates or chains to stop these vehicles.
South Downs National Park Authority confirmed that police have been notified and warned that drivers could be arrested or heavily fined if they damage a scheduled monument, either intentionally or through recklessness.
A spokesperson added: “Kingley Vale is an internationally-important archaeological and ecological site, with 14 scheduled monuments that include a number of ancient burial mounds, as well as rare and important biodiversity.
"The reserve is owned and managed by Natural England and the site manager is aware of this issue.
“The only vehicles allowed on site are those belonging to organisations that help look after the site, including Natural England and Forestry England, as well as the private landowners.
“The track itself is located on private land and the site manager is currently liaising with the landowner to try and find a solution to the issue."
The authority said it continues to work with Natural England and other partners to 'proactively encourage responsible behaviour' at this 'very sensitive site'.
"This ongoing work includes additional interpretation and signage, as well as more on-the-ground engagement with people visiting the reserve," they said.
"Natural England has also appointed a summer warden specifically to help with visitor engagement and monitoring the site.”
Motor vehicles of any kind are not permitted on any right of way other than byways open to all traffic.
Driving on a bridleway or footpath would be trespass against the landowner and in addition is a criminal offence under section 34 of the Road Traffic Act 1988.