Alexys Brown, also known as Lexi, died at her home in Emmadale Close, Weymouth, on Thursday.
An investigation is under way to discover how she became trapped. A post-mortem examination is due to be carried out this week.
It was originally hoped the appeal would raise £2,000.
Alison Record, who started the Just Giving appeal, said she was "heart broken" over the death.
"Everybody by now has heard of the terrible tragedy the Brown family have suffered with the loss of their beautiful and beloved little girl Lexi," the appeal page reads.
"It is such an unimaginable pain to come to terms with; they should be free to take all the time they need to find their way through such dark times with out [sic] having to think about the devastating financial circumstance this will leave them and their children in."
"This has destroyed them; they need time to be with each other so all money raised will go towards giving them time to do this."
Many other comments have been posted on the appeal page.
Steph Harris said: "Thinking of you all at this devastating time, fly high beautiful princess. Love Steph and family xxx"
Lesley Andrews added: "No amount of money will take away the pain, but so much love comes with every penny. Take care. xx"
Aster Group, the housing association responsible for managing the home, is assisting with the police investigation. The Health and Safety Executive (HSE) is also investigating.
Dorset County Council said it had not installed the disabled lift at the property.

Fellow scrum-half Greig Laidlaw was Scotland's man of the match in their opening Pool B win over Japan.
But Pyrgos takes over as skipper as head coach Vern Cotter rotates his squad against the USA on Sunday.
"Greig had a great game on Wednesday, but I'll focus on my role and do it to the best of my ability," said the Glasgow Warriors 26-year-old.
"I just look at it like it's a great opportunity.
"Hopefully, we get the right result."
Pyrgos led Scotland in a World Cup warm-up defeat by Ireland last month and will be determined to retain his place.
However, Laidlaw is almost certain to return for the potentially decisive meeting with South Africa in Newcastle a week on Saturday.
"I knew coming in that was how it was going to go," added Pyrgos.
"It was amazing to lead the team out against Ireland and again it's a huge honour to captain Scotland at a World Cup.
"I know how much it means to the country and I'm really excited.
"I'm not sure how I'll feel when I get out there, but it'll be a big moment for myself and my family."

Farmers, conservationists and the RSPB have worked together to preserve the breeding habitat in fields that has helped the stone curlew survive.
The birds, which winter in Africa, return to Breckland farms in April.
Fields have been specially ploughed for birds to get at worms and insects while plants that attract them line edges.
Those involved in the conservation programme met in Thetford on Friday to celebrate and welcome the arrival back of the birds.
Breeding pair numbers have almost trebled and the RSPB has described the project as pioneering.
Source: RSPB
An online appeal set up to help the family of a five-year-old girl who died when her head became stuck in a lift in Dorset has raised more than £12,000.
[NEXT_CONCEPT]
Henry Pyrgos is ready to make the most of what could be his only chance to captain Scotland at this World Cup.
[NEXT_CONCEPT]
A bird that was on the road to extinction in its native habitat in the Brecklands of Norfolk and Suffolk is thriving after 30 years of care.