The whole question is mentioned in the title but here I would add the conjecture that I am affraid that the answer(s) should be similar to a possible question about Michelson and Morley experiment...
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Yes, there is Doppler shift. There is a transverse Doppler shift because of the tangential motion of the Earth around the Sun. There is also a variable longitudinal Doppler shift caused by the fact that the Earth's orbit is non-circular and there will be a component of the Earth's orbital velocity towards or away from the Sun at different times in the year.
