It works if you add ImplicitRegion:
VectorPlot[{y, -x}, {x, y} \[Element] ImplicitRegion[(1/4 < x^2 + y^2 && x^2 + y^2 < 9), {x, y}]]
You can also use Annulus:
Annulus
VectorPlot[{y, -x}, {x, y} \[Element] Annulus[{0, 0}, {1/2, 3}]]
Works. Great.
