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]