Is it common for inbred groups like ashkenazi jews and the amish to have hereditary problems?

Yes. Many hereditary problems are due to the coming together of two parent 'carriers' of a recessive trait, those traits are more common in those groups due to the closer relationship genetically of these people.
We all carry genes. which may cause disease. Iincreased risk of carrier status for a particular rare disorder is usually from "founder effect"--a small number of people populating an new area and then expanding in population over centuries. French Canadians are the equivalent of 2nd cousins, and we know which rare genes to test for, like Ashkenazi Jews or Cajuns. Their overall rate of genetic disease is not elevated.