Hi and welcome to the forum. If her infection smells and is swollen and red with discharge she could die without antibiotics and cleanup/bandaging. If you can, as has already been suggested, you need to call around and find a rescue/shelter that will be willing to come and set a large, humane trap and pay for care.
Large traps are available for $85 at home depot and they work without hurting the animal. You would just put her food in it and when she went in to feed, the door would shut. Cover it with a blanket to calm her down and she is ready for the car. If shelling out $85 is feasible for the trap then you could also call around to see if any vets would do a good deed and see her for free. If you are near a large university they may have a vet school that could help.
Maybe you could see if a vet. clinic or shelter/rescue could lend you a trap?
I just think that one way or the other she will have to get some antibiotics and get that infection cleaned up. It's even worse that she can't clean it up herself. Truly sad. She is suffering and needs a doc.
Ideally, she would go to a shelter or vet. and get treated. She would have her litter in the shelter and then get spayed so she would not have to go through the same thing again.
I'm sorry you are in such a bad situation and I wish I could do more to help. Good luck, keep us posted if you can.