After a family member's passing, one has the status of "Onen" until the burial. During this period, he is exempt from Misvot, and thus he does not recite Kaddish.
If, however, a family member is not tending to the funeral preparations, then he may recite Kaddish for the parent. Thus, for example, if a loved one passed away on Friday night, or during the night of Yom Tob, one may recite Kaddish, since he is unable to tend to funeral preparations on Shabbat or Yom Tob. Even on a weekday, a person may recite Kaddish for a loved one once the Hebra Kaddisha has assumed responsibility for the burial. This is the ruling of Hacham Ovadia Yosef, based on a responsum of the Noda Bi'yehuda (Rav Yechezkel Landau, Prague, 1713-1793).
Similarly, it is customary for the mourners to recite Kaddish at the funeral after the eulogies, just before the body is taken for burial. Although the family members still have the status of "Onen" at that point – as the deceased has not yet been buried – they are nevertheless allowed to recite Kaddish, and, as mentioned, this is the accepted practice.