Like the radiation the Sun gives out to sustain it's circular shape, will Hawking radiation sustain the black hole's shape?
In a word, no.
A black hole's border or event horizon is just curved space time where the curvature exceeds an escape velocity of the speed of light. Because the inner mass is thought to be very uniform, with essentially zero mass concentrations, it should be one of the most spherical objects in the universe, even more than the sun. A black hole's event horizon's spherical shape has nothing to do with Hawking radiation, which is a quantum effect just outside the event horizon.
That said, a rotating black hole has two distinct event horizons. The inner one is spherical, the outer one is not and that may be a more accurate picture of black holes, so they may not be as spherical as highly spherical stars depending on which event horizon your taking into account.