You might consider SliceVectorPlot3D with "BackPlanes" as the second argument:
e = {Cos[π x/5] Sin[π y / 4], Sin[π x / 5] Cos[π y/4], Sin[π x / 5] Sin[π y / 4]};
SliceVectorPlot3D[e, "BackPlanes", {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]
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