In the ArcGIS.com map viewer, you click the feature once to see the resize handles, then you click the feature again to see the vertices. Now, right click on any vertex to delete it. This works in all browsers, except IE9 right now. I just tried IE7, Safari, Firefox, Chrome and they all work
In Explorer Online, I think you've found a bug. You should be able to hover over a vertex and click the Delete key on your keyboard to delete it. For me, however, it deletes the whole feature, not just the vertex. I'll report this.
Thanks,
Mike