Cork is usually finsihed with wax or a polyurethane finish that also acts as the sealant. Personally, I recommend using several layers of polyurethane on your unfinished cork floors.
I am assuming you plan to remove the existing coat that is peeling off before trying to apply a new coat. To do this, I recommend that you have a professional come and "sand and coat" your floors. You can do this yourself, but be prepared for the cost of equipment rental added to the cost of supplies. You should also consider details such as how you will seal off the room to prevent dust from getting in other rooms and how to protect the wet finish from dust, hairs, etc.
You may be considering just peeling off the existing layer rather than sanding the floors, but I caution against that because it's doubtful that you will get all of it out of the cracks and cervices and when you put the new coats of finish on, it may not stick properly to the areas that still have reminants of the old finish layer.
Let me know if you have any other questions on this!