Hermit crabs most definitely have bowel movements. Their feces is deposited in the shell and they get it out by moving their abdomen around. You may not be able to see the feces because it might blend with the type of material you use in the bottom of your tank.
Yes, your crab should have bowel movements. If they are eating, there will be waste, unless there is a problem such as constipation. Use some common sense to access the problem. Are they eating only one type of food and not a variety that includes some fruit and vegetables? Are they drinking from the water dish that you provide?
Every creature that eats produces fecal matter. If you clean its tank regularly you may not notice it. Think of the shrimp from Finding Nemo. Crustations are dirty like that.
Yes every living thing on this earth has to dispoze of its self one way or another! It might be so small that you cant see it or it might dispoze of it in another way. If a living thing does not dispoze of its toxic chemicals it will die.