Nah. Probably due to an error in measurement. Wouldn't this be fairly simple to confirm or refute by, say, comparing relative fluctuations in neutrino activity originating from the direction of the sun's core and the visible appearance of solar flares, perhaps? A distance of ~90-million+ miles would more than sufficient to produce a noticable gap, methinks.
A radio signal sent from the rear to front of the vehicle would still be clocked at C, no matter what the speed or acceleration of the ship is (including 99.999999999999999999% C).
Doppler shift is a change in frequency, not propagation.