It's very likely the publisher doesn't have the data if it wasn't published as an online supplement to the article.
You should start by contacting the authors (typically by email but perhaps by phone if email doesn't work) and asking for the data. In my experience, many authors will happily give you access to the data, and if they are prepared to do this, a polite request will be the quickest and easiest way to get it. Furthermore, it's most likely the case that you will need help from the authors in understanding details of the data. If you've worked with them in a cooperative fashion, then there's a much better chance that you'll be able to get such questions answered.
There are various reasons why the authors might not be able to share their data with you. For example, in research projects involving human subjects research the IRB approval for the study might not allow for such a release. Another common situation is that a company has sponsored the research and considers the data to be proprietary and the researchers are contractual obligations that do not allow them to release the data.