Food bowls aren't really essential - we all like them because they keep the cage neat
but really I don't think the hamsters mind if their food is in a bowl or scattered all over the cage. In fact scattering might even be better as it provides something for the hamster to do - root around for it's food. I do use food bowls for my own convenience, but often my hamsters have other ideas and throw the food all over the place anyway!
You could try providing a potty like the others have said, and placing it in a corner. Hamsters often like to go to the toilet in a corner, so if the food bowl is in a corner maybe move it somewhere more open.