Frank's neighbor quickly jumped in his car and screeched off toward the wounded man.
Yes, "screeched off" is an idiomatic and fairly common way of describing a car making a sudden start. It refers to the sound of the wheels spinning, trying to gain traction. However, there are a couple of issues with your use of it.
Firstly, you're not saying the car screeched - as you've written it, you are saying the neighbour screeched (he jumped in his car and screeched off), which is a bit weird. I'd understand what you meant, but if you're writing a novel you'd be open to criticism. You could attribute the sound to the car by instead saying:
Frank's neighbour quickly jumped in his car which screeched off toward the wounded man.
The other thing that strikes me as odd is that, if the driver can see the wounded man, how far does the car have to travel? From a logical point of view it seems odd to me that someone would slam their accelerator pedal down, spinning the wheels of their car to get a short distance up the road, not to mention dangerous - does he want to run over the wounded man?
I'd consider saying instead:
Frank's neighbour quickly jumped in his car and sped toward the wounded man.
"Sped" can be attributed to either the car or the driver - they are both travelling with speed. It simply implies he used his car to get to the wounded man as quickly as possible.