Since taking over on 1 April, the company has been plagued by delays with numerous patients missing appointments.
Coperforma apologised and on Thursday evening confirmed that one of the senior managers was leaving his post.
It had previously said the problems were due to "inaccurate data" being transferred to the company.
Gary Palmer, from the GMB union, said the blame rests with the High Weald, Lewes and Havens Clinical Commissioning Group.
"The service was successful, it had 90% satisfaction levels," he said.
"If their aim was to seek to improve that somewhat, they've certainly done as much as they possibly could to actually destroy service provision within Sussex."
Dialysis and cancer patient Lois Wood said she had missed five hospital appointments.
"I was frightened as well as cross," she said.
"I have been left at the [dialysis] unit at 10 at night four or five times. Brighton Hospital has had to get emergency taxis for us."
It was revealed on Thursday that Stephen Payne, the business manager at Coperforma, will leave his post at the end of the month.
The High Weald, Lewes and Havens CCG said the process of hiring Coperforma was being investigated independently.
They said the report is expected in June and the group will not comment until then.
Thousands of people have signed a petition calling on the NHS to end its contract with a company recently chosen to run patient transport in Sussex.