When the light reflects off the wall, it will act as if the wall is a new source for the light. If the way itself is not moving, it will not Doppler shift the light. But if the siren is rotating, the motion of the siren will Doppler shift the light. Reflection off the wall will retain that shift, but won't add to or subtract from it.

On the other hand, if the "wall" is an atom or an electron, and it moves or scatters when the light reflects off of it, then that will cause an additional Doppler shift.

When the light reflects off the wall, it will act as if the wall is a new source for the light. If the wall itself is not moving, it will not Doppler shift the light. But if the siren is rotating, the motion of the siren might Doppler shift the light, depending on how the light source is mounted to the siren.. Reflection off the wall will retain that shift, but won't add to or subtract from it.

EDIT: Based on your comments, I may have misunderstood the situation you are describing. If the light is pointed radially outward from the center of the siren, and the siren is rotating, then the instantaneous velocity of the light source is perpendicular to the direction of propagation of the light. In that case, there will be no Doppler shift. The reflected beam would also have no Doppler shift.

Regardless of how exactly the light source is moving or rotating, if the "wall" is an atom or an electron, and it moves or scatters when the light reflects off of it, then that will cause an additional Doppler shift.