The Volkswagen Golf smashed into the house in Morehall Close, Clifton, in the early hours of August 2 and then caught fire.
Now family friend Lauren Purvis has set up an appeal on the JustGiving website with the hope of raising £5,000.
She said her friend’s stepfather, who was in the house at the time, was having a second operation.
Writing on the fundraising page, Ms Purvis explained: “They have had to move out of their family home. The fire created by the crash has destroyed all their belongings. My friend’s stepdad is currently in hospital having his second surgery.
“We would really appreciate people’s donations to help this family at this tragic time.”
A man in his 20s, believed to be the driver, was arrested.