The error message that
VectorPlot3D[{x, y, z}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1},
VectorPoints -> {1, 1, 1}]
produces
suggests that two is indeed the minimum number of VectorPoints.
EDIT: Using
VectorPlot3D[vp1=VectorPlot3D[{x, y, z}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1},
VectorPoints -> {2, 2, 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]]
