If you use an unmanaged switch to a WAP that uses VLANs, you have no idea what will happen until you do. There is no standard for what an unmanaged switch will do with a VLAN tagged frame. Some unmanaged switches will drop the frame as damaged, some will strip off the VLAN tag and forward the frame, and some will simply pass the frame as is. Two out of the three options are bad, and you will not know until you try it.
A PoE injector will not look at the frames, and it will simply pass on all the layer-1 signals it receives.