If you have a problem with scaling you must add "Plot range".

f[x_, y_, z_, g_, m_, v_] :=
 {((x)^2 - m), x*y, x*z}*(g*v^2)/(2*(g^2*((x)^2 - m)^2 + (v*y)^2 + (v*z)^2)^(3/2));

 Animate[VectorPlot3D[f[x, y, z, 1, m, 1], {x, -5, 5}, {y, -5, 5}, {z, -5, 5},VectorColorFunction -> Hue, PlotRange -> 5], {m, -5, 5, 1},AnimationRunning -> False]

Export to Animation.

ANIM = Table[
   VectorPlot3D[
    f[x, y, z, 1, m, 1], {x, -5, 5}, {y, -5, 5}, {z, -5, 5}, 
    VectorColorFunction -> Hue, PlotRange -> 5], {m, -5, 5, 0.25}];
Export["Test.gif", ANIM, ImageResolution -> 200]

In Documentation Center is written:

With the vector scaling function set to None, then all vectors have the same size.

Animate[VectorPlot3D[
 f[x, y, z, 1, m, 1], {x, -5, 5}, {y, -5, 5}, {z, -5, 5}, 
 VectorColorFunction -> Hue, PlotRange -> 5, 
 VectorScale -> {Small, Automatic, None}], {m, -5, 5, 0.5}, 
AnimationRunning -> False]

or

Animate[VectorPlot3D[
  f[x, y, z, 1, m, 1], {x, -5, 5}, {y, -5, 5}, {z, -5, 5}, 
  VectorColorFunction -> Hue, PlotRange -> 5, 
  VectorScale -> {Small, Small, None}], {m, -5, 5, 0.5}, 
 AnimationRunning -> False]

I hope it will help.