Frank, I just want to mention that the links Tim Witt is pointing you to are for the "developers" version of the Web App Builder (WAB), not the ArcGIS Online (AGOL) version. I only point that out because of the location you posted the question (ArcGIS online).
WIth that said, what Tim mentioned is correct, no ArcGIS Server is required. An account on ArcGIS Online, or a local Portal (basically the local equivalent to AGOL, which does come with some versions of ArcGIS Server) is required to register the WAB. You can then pull everything you want, service wise, from AGOL. This may be a little different from downloading what you've customized using a template on AGOL and then downloaded, so keep that in mind (If that is what you where specifically asking about).
The WAB-dev developers' community is very active, however, so check out Web AppBuilder Developer Edition – Customization Resource List specifically to see a list of the many custom widgets users have already created.
also, just fyi, updates and new features (from esri) typically are released first to AGOL, a bit later to the WAB-dev, then to Portal. So what you see as a widget on AGOL today, might not be available quite yet with out-of-the-box WAB-dev..
