Yes, if he wishes to do so he may certainly date while in the discernment process. Dating may even help him to discern whether he is being called to marriage instead of the priesthood, and it may help him to build healthy chaste friendships with women (something that is important for priests to have). However, he should not be invited and/or pressured to date by someone else (who should respect his possible call to celibate chastity); and he should stop dating once he has committed himself to applying for and training for the priesthood.
