In his contract a customer refers to the J-STD-001JS. Is this a mistake in writing, since the document we use is designated J-STD-001J?

No, apparently this customer is related to the space industry. For this group of products IPC has developed a so called addendum, a document containing supplementary specific criteria. This means that this product is subject to the criteria in the regular J-STD-001J as well as J-STD-001JS (mind the priority however).