He doesn't have a shower in that scene. Shortly after being rescued, he comments that he hasn't had a shower in some considerable time, which explains his ripe odour.
Martinez: [Eww]There a little smell going on there, bud...
Watney: [laughing] Well, I haven't had a shower in a year and a half!
He appear to be dry-shaving in the MDV (MAV) rather than the Rover. He also seems to have a number of items around him that he's preparing to jettison. One assumes his shaving kit will be leaving via the porthole shortly, to save weight.
This is indirectly mentioned in the novel
After Johanssen gave us the all clear, Dr. Bossy-Beck made me wait
while he first took off his suit, then took off mine. After he pulled
my helmet off, he looked shocked. I thought maybe I had a major head
wound or something, but it turns out it was the smell. It’s been a
while since I washed…anything.