An object moving and an object vanishing are the same from the perspective of wave propagation, the only difference is that one event will have a more dramatic (therefore easier to visualize) effect.
If the sun instantaneously vanished, we would see it disappear at the same instant as its gravitational effect stops, 8 minutes after the actual event occurred. For those 8 minutes while the light and gravitational information are in transit, the Earth will continue to revolve around a visible (though now nonexistant) sun.
In the same way as if the sun suddenly jerked ten million miles to the south, we would see it move at the same instant as its gravitational force vector changed, 8 minutes after the actual event occurred, but that's harder to keep in your head.
If the sun instantaneously vanished, we would see it disappear at the same instant as its gravitational effect stops, 8 minutes after the actual event occurred. For those 8 minutes while the light and gravitational information are in transit, the Earth will continue to revolve around a visible (though now nonexistant) sun.
In the same way as if the sun suddenly jerked ten million miles to the south, we would see it move at the same instant as its gravitational force vector changed, 8 minutes after the actual event occurred, but that's harder to keep in your head.