Is this documentation simply out of date? Is it possible to manipulate vertex weights in Edit Mode in Blender 2.8?
It looks like you have the exact same thing in 2.8, but to see the weight for the selected vertex you first need to assign it some weight, either in Weight Paint mode or with the Assign button under the Properties panel > Object Data > Vertex Groups list. As Robin says: The panel is displayed only when you have a vertex which has been assigned to a vertex group selected, even with a 0 weight value.
To see both the wireframe and the weight you can enable Vertex Group Weights in the Viewport Overlays.