This is a rather tricky situation, and it is difficult to give a clear answer without knowing who actually owns the samples and whether Researcher A has the authority to give the samples to Researcher C. I think the problem is between Researcher A and Researcher B. If Researcher A does not have the authority to give the samples to Researcher C, then Researcher A might land up in trouble.
However, as far as Researcher C is concerned, if she has evidence that the samples were given to her by Researcher A, I don't think she needs any permission or authorization by Researcher A or B to publish her research. However, she should acknowledge that Researcher A has helped her by providing the samples. In fact, it might be a good idea to acknowledge both Researcher A and B and the Research Centers X and Y. A few sentences saying that the samples for the study were provided by them in the acknowledgments section would suffice for that.
The International Committee of Medical Journal Editors (ICMJE) has provided a set of guidelines meant to help researchers decide whether a contributor qualifies to be an author or should be just be included in the acknowledgements section of a manuscript. According to these guidelines, "Those whose contributions do not justify authorship may be acknowledged individually or together as a group under a single heading (e.g. "Clinical Investigators" or "Participating Investigators"), and their contributions should be specified (e.g., "served as scientific advisors," "critically reviewed the study proposal," "collected data," "provided and cared for study patients", "participated in writing or technical editing of the manuscript")."