The error message that
   
    VectorPlot3D[{x, y, z}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, VectorPoints -> {1, 1, 1}]

produces

![enter image description here][1]

suggests that  two is indeed the minimum number of `VectorPoints`.

EDIT: Using 

     vp1=VectorPlot3D[{x, y, z}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, VectorPoints -> {2, 2, 2}]

![enter image description here][2]

a workaround is possible by manipulating the components of the `Graphics3D` object `vp1`:

    GraphicsGrid[
    Partition[
    (vp2 = vp1; vp2[[1, 2, 1, 2]] = vp2[[1, 2, 1, 2, #]];vp2) & /@ Range[8],
    4]]

![enter image description here][3]


  [1]: https://i.sstatic.net/B1juP.png
  [2]: https://i.sstatic.net/8srz8.png
  [3]: https://i.sstatic.net/3ygrw.png