Wigs can only be obtained by Raiden on the Plant mission.
In order to obtain the orange wig, you need 170 dog tags (70% of the total amount of dog tags) on the plant.
Dog tags can be carried over from one difficulty to the next IF you use the same save. For example: If you beat the plant on Easy, save it when prompted after the end cutscenes and credits, and then load that save you can play on higher difficulties (i.e. Normal, Hard, etc).
If you want to see how many dog tags you have collected, go under the "Special" option in the title menu, and select "Dog Tag Viewer". Dog tags collected will give you the location, so you can somewhat figure out where to find the missing ones.
Questions, corrections and neg vote justifications below.