Tower Hamlets Council has since cancelled the fine and hand-delivered a note to the family's home to apologise.
Andre Spicer, a New Zealander with an Australian partner, recently tweeted: "We have been overwhelmed by the kind response from people across the world.
"Dozens of festivals, markets and businesses have offered us the opportunity to set up a lemonade stand."
Among those extending the offers was London's Borough market which tweeted the girl's father:
Lee Fest: Neverland has also invited the little girl to sell lemonade at its August festival in Kent.
While Spicer's daughter is not currently too keen on another lemonade stall, even with a permit, he wants to encourage other parents to get their children outside.
"As all childhood research tells us, often they learn through doing things and getting out there," he said.
"Doing something is often far better than any other kind of learning experience."