Arriva Transport Solutions, which carries non-emergency patients in Leicestershire and Rutland, was fined nearly £45,000 last month.
The family of a man who was left waiting nine hours for a hospital transfer after a stroke said it should not be allowed to renew its contract.
The local NHS group said most patients were happy with the service.
Arriva boss Paul Willetts said "it was not good enough" and apologised for missing targets for getting patients to and from hospitals on time, which he said was partly down to increased demand.
He also apologised to the family of Michael Burden, 76, but his daughter Emma Purcell said the company should not be "rewarded" by having its contract renewed.
"When they're consistently letting people down, why would they get the contract again?" she said.
West Leicestershire Clinical Commissioning Group, which manages the contract, has fined Arriva in each of the three years its contract has been running.
The group's chief executive Toby Sanders said 88% of patients would recommend the service to family and friends.
"We don't always get it right and Arriva have recognised that... We have used fines as part of our contract but actually it's not the fines that improve outcomes for patients," he said.
"What we've been focusing on is improving the service, improving the response times and also making sure we've got the right connections with the other parts of the service."
He said there were 18 months left on the contract and it "would be a poor use of public money" to terminate the agreement before then.
A private ambulance provider has apologised for delays after missing targets for three years running.