`Export` to `"VTK"` does not handle `Arrow`, not even `Arrow[Tube[..]]`; but it can handle `Tube` and `Cylinder`:

    $maxlen = 0.08;
    $aratio = 4.;
    tubeArrow[p : {p1_, p2_}, radius_: 0.05] := Module[{len, v, r},
       v = p2 - p1;
       len = 0.5 Norm[v];
       len = Min[$maxlen, len];
       v = len*Normalize[v];
       r = len/$aratio;
       {Tube[{p1, p2 - v}, 0.5 r], Cone[{p2 - v, p2}, r]}
       ];
    
    VectorPlot3D[{x, y, z}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}] /. 
     Arrow -> tubeArrow

[![enter image description here][1]][1]
    
    Export["/tmp/foo.vtk", 
     foo = VectorPlot3D[{x, y, z}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}] /. 
       Arrow -> tubeArrow]
    (*  "/tmp/foo.vtk"  *)
    
    Import["/tmp/foo.vtk"]

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/FMxun.png
  [2]: https://i.sstatic.net/Y8Hbu.png