In what ways does the story "A Good Man is Hard to Find" use the grotesque to show that without God there is no value system?

In literature, the grotesque refers to the different and various ways the human body can be distorted or exaggerated. Flannery O'Connor used this literary technique in many of her short stories and novels. In her short story "A Good Man is Hard to Find," from the anthology of the same name, there are several ways in which she uses the grotesque.
From a strictly physical viewpoint, the mother's broken shoulder hangs helplessly and grotesquely at her side after the car accident. The grandmother dies with her legs folded under her in a childlike pose and with a smile on her face.
A character can be considered grotesque even if they evoke in the reader both empathy as well as horror. The Misfit, by this definition, is a grotesque character. He is very mannerly and seems kind. He's certainly more respectful than the spoiled grandchildren. However, regardless of his demeanor, he is responsible for the murder of five innocent people. Consider this quote from the Misfit:

"Nome, I ain't a good man," The Misfit said after a second as if he had considered her statement carefully, "but I ain't the worst in the world neither. My daddy said I was a different breed of dog from my brothers and sisters. 'You know,' Daddy said, 'it's some that can live their whole life without asking about it and it's others that has to know why it is, and this boy is one of the latters. He's going to be in to everything!'" He put on his black hat and looked up suddenly and then away deep into the woods as if he was embarrassed again. "I'm sorry I don't have on a shirt before you ladies," he said, hunching his shoulders slightly.

Here, O'Connor shows the serial killer in a sympathetic light. He has manners and is concerned about offending the grandmother and her granddaughter. O'Connor also creates sympathy for this character when she describes his life and the many things he accomplished. People don't see serial killers as humans so much as monsters. We usually only know of their crimes, not their talents and interests. Another way she creates sympathy for this character is that he has been punished for something he doesn't even remember doing, killing his father whom he describes as having "a heart of gold." Careful readers will also note that he did seem to only want to help the family get their car back up and running at first. By his own admission, he wouldn't have killed the family if the grandmother hadn't recognized him. The irony is that as readers develop sympathy for this character, he callously murders or orders the murders of five innocent people.
Hiram can also be seen as a grotesque character in the juxtaposition between the way he tenderly helps the mother up and to the woods before shooting her. He also tenderly assists Bailey to the woods before shooting him. O'Connor describes it as the way one would assist an old man. Bobby Lee is similarly grotesque, though less sympathetic. He doesn't seem to initiate any kindnesses but performs them at the request of the Misfit. He checks the car out to see if it can be fixed and holds June Star's hand—or tries to—as he is leading her to the woods. At the end of the story, he boasts about the fun he is having before the Misfit tells him to shut up.
In many ways, the grandmother is a grotesque in the opposite way that the Misfit is. She is not someone readers have empathy for. She is bent on getting her own way, even lying about the secret panels in the mansion to get the kids to pester their father to take the fatal route. She is mostly concerned with appearances but not authenticity, and knows the words to say—like telling the Misfit to pray and ask Jesus for help, but doesn't couple the words with the love that the Misfit so desperately needs. Not until the moment just before her death does she extend grace and love to the man. So, she begins as an unsympathetic character but ends up an authentic, sympathetic character showing love to the unlovable.
So to address the part of your question that asks how O'Connor uses the grotesque to point out that without God, there is no value system, the Misfit knew about God but did not seek a relationship with Him or religion. He felt that he wasn't the worst of humanity, and his crimes were simply things that had to be done. He thought that the punishment of his initial crimes was long and unusual. This points out that man is not fit to judge what is wrong and what is right, because while murder is heinous to the victims, it is often justifiable by the perpetrator.