The Oort Cloud
The far edge of the Oort Cloud is believed to be 100,000 AU from the Sun. In fact, Duncan et al. (1987) created simulations that showed the inner edge of the Cloud to be at ~3,000 AU.
Given that 1 AU $\approx$ 8 light-minutes, $$3,000\text{ AU}\times\frac{8\text{ light-minutes}}{\text{AU}}=24,000\text{ light-minutes}\gg1\text{ light-day}$$ Therefore, any signals - even from the inner edge of the Cloud - will not reach Earth for far more than one day, at which point the ship will be long gone.
Furthermore, any signals - even gamma-rays - should be reduced in strength, compared to signals from areas near any of the planets, the Sun, or the asteroid belt.