Yes it has doppler shift applied, obviously at normal speeds its very very low and you can't measure it. From a fast moving sun in another galaxy it's very easy to detect and you know it as cosmological red shift. There is a slight hand wave in that answer in that classical doppler differs slightly from relativistic doppler but for the example you are using they will both give the same answer. However I suspect you are talking about the microscopic doppler of you standing near the atom.
Now if you were really thinking you would realize there is also a recoil of the atom as it emits the photon. As the atom is charged it must therefore produced a thermal emission and it does and has a name of recoil temperature or sometimes photon recoil heating
https://en.wikipedia.org/wiki/Recoil_temperatureAs given in the article it's the lower limit for laser cooling and is very small under 1uK (1 micro Kelvin), you need tricks to go lower.
Small as the recoil may be it has been directly measured many hundreds of different ways in efforts to push laser cooling as it had to be understood correctly. Google "recoil energy direct measurement" will get you any number of the experiments.
You may get some interest looking at the machines used which are called reaction microscopes. They can do single photons as well as other more complex kinematic processes
Reaction microscope:
https://www.mpi-hd.mpg.de/pfeifer/page.php?id=126So the orbital emission case that everyone on the forum seems hell bent on attributing everything too also makes a thermal emission which will catch them out completely. So when we say orbital emission is only one frequency we overlook this based really on the magnitude difference between the two emissions much like we often overlook friction it depends with how much accuracy you want to apply to the discussion. At this limit we really are splitting hairs these emission play no real role in anything that isn't down near the quantum limit.
