Possibly, if the economics warrant it. I say this because there is still plenty of uranium left in the ground since the uranium mining industry folded in the USA. I assume at some point in time when mines become depleted or uneconomic we will need to reprocess.
Regarding reprocessing, while you may store the waste in a repository you can still pull it out again to reprocess it. The advantage, presumably, of having it in a centralized repository is to protect it from terrorists as well as the environment. It would take many years to fill a repository and the disposal would not be finalized by sealing it, I assume, until a decision was made to reprocess or not. But, if the government and envirowackos are in the loop, I have small hope it will be a decision based on science.