You do not need to know Reiki to learn and practice Core Healing. When you learn to identify the root cause of a persons energy block, and then learn how to clear it out of their chakras, a space will be left. To help ensure that the energy block does not rubber band back into the person (which can happen if they are not willing to let it go, no matter what you do) you will learn to fill that space with White Light. Anyone can do this.
If you take any Psychic Developement classes, you will learn to meditate and invoke the White Light of God. It can be used in many ways, in Core Healing, we use it to fill the space left by the energy block that we have cleared out.
For practitioners who do know Reiki, you can use your Reiki skills to fill that space with the healing energies you have been attuned to channel.
Personally, I feel the results are best when Core Healers and Reiki practitioners work together. One clears and the other immediately fills the person with healing energy. However, a person who knows both can do it alone as well. Also, a person who knows only Core healing can fill the space with white and even golden light.
So, while not required, if you do know and/or practice Reiki, this would be a wonderful way to use your gift of healing.