Is it possible to get 50 50 physical custody without either parties paying child support? I am currently unemployed and my wife is working at walmart. She is persistent on a divorce, even though I was willing to do whatever it takes to save our marriage. Is it maybe in my favor to receive child support from her? Or would a situation like this ever go in the father's favor. My child is only 13 months.

It's possible to get 50/50 custody and neither parent pays the other.
You won't know until the court decides. A lawyer might give you an idea.
Somehow I doubt that you would have a chance of getting support from her on her Walmart salary, just because you are unemployed and didn't want the divorce.
This sounds catty, but you do say you want to save the marriage. Could your lack of work have anything to do with her wanting a divorce? Where I live, Home Depot and Comcast are always hiring. Probably about the same wages as Walmart, but hey. Meanwhile, is the baby diapered and fed and the house clean and laundry done and dinner on the table when she gets home from work?

First ANY question on law needs to include your general locale as laws vary by area. But it is certainly possible to have an arrangement of alternate weeks. That would involve no child support. This generally only works while the child is of preschool age and/or if the parents live in the same school district. Once the child goes to school, they have to remain in the same school district.

Generally the courts do favor the mother for primary custody, but it depends on local court sentiments. A local lawyer will be able to advise you better on this.

You will have a very hard time substantiating why you want custody if you have to job to support your child.