Yes, it is entirely possible. This is why emission bands in real life have a spectral width to them; it is because of the doppler shift of the light that is emitted from moving ions. Light emitted from an ion that is moving towards you will have a slightly higher frequency than light emitted from an ion moving away. This effect is called Doppler Broadening.
Edit: Ok, I misunderstood the original question. If the electron is moving before being captured by the ion, then this will also change the energy of the emitted photon as the OP suggested. The emitted photon is the difference in energy between the electron before capture and after capture, so if the electron has some initial kinetic energy then there is a bigger energy difference between that energy state and the lowest energy state of the atom as compared to the difference between an electron at rest and the lowest energy state of the atom. Of course, it's possible that the electron will graze by the ion on the first couple passes and lose energy gradually through emitting lower frequency photons, so that by the time it's captured its kinetic energy is close to zero. Also, if the electron's kinetic energy is greater than the potential well of the ion, then the electron will just be scattered rather than captured.