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This answer is similar to @Edmund's, except with the rotation angle dependent on the radius:

pts2 = Table[
   Table[{x Sin[t], x Cos[t]}, {t, 0, 2 \[Pi], \[Pi]/(2 x)}], {x, 1, 8}];
a = Table[VectorPlot[{vx[x, y], vy[x, y]}, {x, -2 \[Pi], 2 \[Pi]}, {y, -2 \[Pi], 2 \[Pi]}, 
    VectorPoints -> (Flatten[Table[RotationMatrix[\[Phi]/x].# & /@ pts2[[x]], {x, 1, 8}], 1])], {\[Phi], 0, 2 \[Pi], \[Pi]/20}];

Export["vectors.gif", a]