The group has teamed up with legal firm Thompsons Solicitors, a practice which specialises in personal injury cases, to call for the investigation.
The recent legionella outbreak, which killed three and left dozens more seriously ill, was declared over on Wednesday after health bosses established that no new cases had emerged outwith a 14-day incubation period.
An investigation into the source of the outbreak by health and city council leaders is continuing. Police and prosecutors are also examining the three deaths.
However, campaigners today attacked the announcement as premature and said only a public inquiry could establish what caused the outbreak and whether it could be considered over.
Campaign member and mother-of-three Helen Booth, 59, who was admitted to Edinburgh Royal Infirmary with the bug last month, said: “I really think that someone needs to be punished for what has happened and questions need to be answered. We have every right to be able to walk through our town in a safe environment and breathe clean air.
“This is not about money. The families who have lost their fathers and sons – I am fighting for them. I feel that there’s been a big cover-up somewhere.
“The public deserve to know what’s going on and they need to have their say.”
Mrs Booth, a receptionist from Clovenstone, added: “I was gutted when they declared the outbreak over. I come into work every day and I think, ‘hell’s teeth, am I going to get this disease again?’”
Thompsons said the outbreak highlighted weaknesses in the enforcement of health and safety regulation at work.
Patrick McGuire, a partner at the firm, said: “Considering the serious threat an outbreak of legionnaires’ disease poses, it’s frightening to discover that the laws and regulations relating to the risk assessment of workplaces are woefully insufficient and allow employers to cut corners if they so decide.
“The public are in being put in danger and there’s no safety net. This cannot be allowed to continue. Lives are at stake.
“After all the official meetings at Holyrood we’re still no further forward. We fully support the calls for an independent public inquiry.”
Dr Alison McCallum, director of public health and health policy at NHS Lothian, said: “All the evidence suggests that the actions taken by the incident management team were timely and effective and minimised the impact of this outbreak on the public health.
“Laboratory work is ongoing to try and identify the source conclusively but, as we have said from the start, it may not be possible to do so.
“NHS Lothian would participate fully in a public inquiry should there be one.”