An NHS spokesperson said: "We are looking into the concerns raised about Medefer and will take further action if appropriate."
Medefer's system allows patients to book virtual appointments with doctors, and gives those clinicians access to the appropriate patient data.
However, the software bug, discovered in November, made Medefer's internal patient record system vulnerable to hackers, the engineer said.
The software engineer, who does not want to be named, was shocked by what he uncovered.
"When I found it, I just thought 'no, it can't be'."
The problem was in bits of software called APIs (application programming interfaces), which allow different computer systems to talk to each other.
The engineer says that at Medefer those APIs were not properly secured, and could potentially have been accessed by outsiders, who would have been able to see patient information.
He said it was unlikely that patient information was taken from Medefer, but that without a full investigation, the company could not have known for sure.
"I've worked in organisations where, if something like this happened, the whole system would be taken down immediately," he said.
On discovering the flaw the engineer told the company that an external cybersecurity expert should be brought in to investigate the problem, which he says the company did not do.
Medefer says the external security agency has confirmed that it has found no evidence of any breach of data and that all the company's data systems were currently secure.
It says the process of investigating and fixing the API flaw was "extremely open".
Medefer said it had reported the issue to the ICO (Information Commissioner's Office) and the CQC (Care Quality Commission), "in the interests of transparency", and that the ICO had confirmed there is no further action to be taken as there is no evidence of a breach.
The engineer, who had been contracted in October to test for flaws in the company's software, left the company in January.