Ashleigh Thornhill from Gainsborough had just returned to her home in Springs Road, Gainsborough, from hospital with her newborn when the incident took place at midnight on Monday, April 29.
Ashleigh Thornhill has asked people to share her Facebook post to try and find out who did this.
She wrote: “If anybody has any information on who decided to throw a petrol bomb into my car whilst we slept inside with our newborn baby, who we had only just arrived home with, it would be much appreciated.
“The police have lots of evidence and I’m pretty sure somebody will know who it was, as it was clearly a targeted attack as cars parked either side of mine were left untouched.
“After not sleeping for days and just having given birth it’s fair to say I will not just let this go and will find out who it was.
“Several neighbours witnessed it also and there was somebody on a moped.
“The police have said whoever did it would have been quite badly burnt by the looks of it.”
Firefighters from Gainsborough attended Park Springs Road, Gainsborough to the vehicle fire.
The crews used two hose reels and one dry powder extinguisher to put it out.
A spokesman from Lincolnshire Police said: “At midnight on Monday, April 29, we received a report of arson in Springs Road, Gainsborough.
“This was in relation to a car fire.
“We are treating this as suspicious and investigations are ongoing.
“Anyone with any information should call us on 101, quoting incident number one of April 29.”