The document, signed by 16,000 people, was delivered to Ms May's Downing Street office inside a green box with Grenfell written inside a heart.
It calls for a panel from diverse backgrounds to assist in the probe into the fire that gutted the 24-storey west London tower in the early hours of June 14.
Retired judge Martin Moore-Bick is currently leading the public inquiry, which began holding procedural hearings on Monday, but May has powers to opt for a panel-led inquiry, rather than relying on one chairman.
"Sir Martin Moore-Bick is an experienced High Court judge, who we feel is suited to the Grenfell Tower inquiry, however we're worried that he may not have an understanding of the wider issues beyond his experience and expertise," Sandra Ruiz, who lost her niece in the fire, said outside Downing Street.
"Nor is it right that the truth surrounding the deaths of 71 victims be dependent on a single person's judgement.
"What we seek is an impartial and independent decision-making panel, with a range of experiences to assist the chair -- two, three or four heads are better than one," she added.
Ruiz was flanked by two other relatives and a survivor as she delivered the petition, which has been signed by singer Adele.
"We hope the prime minister is listening and agrees to what are reasonable and proportionate requests," said Ruiz, adding it would be a "tragedy" if their requests went unheeded.
The hearing on Monday ruled that 424 individuals or groups, including some survivors, would be granted "core participant" status in the inquiry meaning they have access to documents, can make statements and can question witnesses through legal representatives.
But Karim Mussilhy, who lost his uncle in the fire, said this was not enough.
"We're really losing confidence in this public inquiry and we're really losing confidence in everything to do with Grenfell -- we feel like we're being forgotten and not listened to," he said in a message to the prime minister.
