Capita took on the national contract from the NHS for delivering patients' records, when people move from one GP to another, in March.
The survey of 78 GP practices shows 9,009 records have been missing for more than two months.
But, Capita said it did not "recognise these claims".
"[However] we know there have been serious issues with services delivered by Capita which have had an unacceptable impact on practices," a spokesman for NHS England said.
"We are ensuring Capita takes urgent steps to improve services."
Paul Conroy, a practice manager at the Mersea Island Medical Practice in Essex, has started a House of Commons petition on the delays, which has been signed by more than 3,000 people.
"GPs rely on that full medical history in order to make key clinical decisions on patient care," he said.
"If they can't get hold of that physical record there could be vital information there could be vital information that puts a patient at risk."
James Dillon, director of Practice Index - an organisation bringing together practice managers - said: "GP practices are getting more and more frustrated by the missing patient records.
"Not only is this debacle putting the health of their patients at risk, it is putting added pressure on already stretched practices."
Dr Ian Hume, a Diss GP who represents the British Medical Association in Norfolk and Suffolk, said: "Delays make the system more cumbersome."
In a statement, Capita said it had taken on the "challenging initiative" to streamline GP support services and there had been "teething problems".
"[But] medical records are now being delivered securely up to three times faster than under the previous system," it said.
"We do not recognise these claims regarding thousands of files being missing whatsoever.
"We request and move on average 100,000 files a week from multiple sites including GP surgeries and also third party run storage facilities which are contracted and managed by NHS England."
More than 9,000 patients' records in Norfolk, Suffolk and Essex have gone missing since a private firm took on transferring files, a BBC survey shows.