Lorraine Lyons, 48, who runs the Charity Search shop in Station Road, said she has listened to dozens of "horror stories" following changes to the way appointments are made at Shirehampton Group Practice health centre.
The stories have led her and her assistant Janet Long to start a petition which they plan to send to the local primary care trust.
Mrs Lyons said: "I have listened to so many sad stories from vulnerable people. The old system worked, you could see your doctor within a week either by making an appointment or ringing on the day they were on duty.
"Now you have to wait weeks. When I called up I was given a date to see my usual doctor and it was seven weeks away, so I said I would see someone else."
She added the system, which involves a triage GP calling patients to diagnose them and arranging for them to come into the surgery if necessary, is also flawed for people who work and "may not have the privacy to talk".
Mrs Lyons and Ms Long have gathered 136 signatures on the petition just from people coming into her shop to complain.
She said: "We've had people coming here in tears, stories like one lady with cancer was given an appointment six weeks away, one man who had to be rushed to hospital while he was waiting for an appointment and one lady who could not get an appointment for her husband who had just came out of hospital after a stroke.
"Millions of pounds have been spent on a new health centre, which seems to be a complete waste of space if you cannot see a doctor. Something has to be done, before someone dies."
While the practice could not comment on individual cases they said appointments are not normally booked more than four weeks in advance.
Jude Ward, practice manager for Shirehampton Group Practice, said: "We are sorry there have been some problems with the availability of routine appointments and that it may be difficult for patients to see a GP of their choice. This problem has been exacerbated by recent illness and maternity leave affecting several GPs.
"Patients requesting urgent or emergency appointments have always been and continue to be dealt with on the same day. Due to feedback from patients, we have made several changes to the new appointment system. The outcome will be monitored and further changes made as necessary."
She said the triage system was part of a national practice following outbreaks such as swine flu.
She added: "The triage GP has access to the patients' medical records and any hospital letters. The GP may then give advice, arrange for a prescription or test or allocate an appointment that day."