Richard Griffin and William Hammersley, 79, from Chesterton, died in 2012. Harry Cadman, 71, from Stoke-on-Trent, died from a pre-exisiting condition.
JTF admitted failing to ensure the safety of employees and customers under the Health and Safety at Work Act at Stafford Cown Court on 30 June.
In a joint statement, relatives of the men said: "The past five years have been extremely difficult, we have had to endure years of not knowing who was responsible for the outbreak of Legionnaires' Disease at the JTF store in Fenton that had such tragic consequences for us all.
"JTF have hid behind a wall of silence all these years before finally admitting that they were to blame, not only for infecting all those innocent people but to causing the premature deaths of our loved ones."
JTF said: "The business apologises unreservedly for the failings back then that had such devastating consequences.
"Since 2012, a completely new board of directors and senior management team have been appointed and tasked with undertaking a continuous and thorough review of the way in which the company manages its business; in particular, health and safety.
"A rigorous and extensive review of the company’s health and safety controls, management procedures and risk assessments has been completed.
"The business considers the health and safety of its customers and employees as its top priority at all times. The company is dedicated and committed to investing in the training of its employees and fostering a positive health and safety culture across the business.
"Again, JTF wishes to extend its sympathy and deepest regret for the events which occurred in 2012 and to all those affected".