Is it permissible upon waking directly from sleep to open one's hands for supplication or to anoint one's mouth, without first washing them?

There is no harm in raising one's hands in supplication or applying oil after waking up from sleep. The fact that you do not know where your hand rested does not prevent this. The majority of scholars consider it disliked to dip one's hand into a vessel before washing it three times when there is doubt about its purity, whether that is after sleeping at night or during the day. This prohibition does not include placing the hand in oil.