Yes, but with a caveat.
Reaper (and Mei, to cover both immunities) can both contest points or payloads owned by the enemy. This is easily seen in most games, so I don't feel proof is necessary.
The caveat here, is that they cannot capture points. I did not test payloads, but I can safely assume the same logic is behind it.
Reaper using Wraith form on an unconstested (neutral) point.
Mei using Ice Block on an uncontested (neutral) point.
If necessary, I can also knock up some video evidence of trying to capture an enemy point while not contested, but I don't think there will be a difference.
As per Blue-Raja's comment, I did further testing on the nature of Overtime and contesting versus capping. My results were fully in-line with the above evidence- both can contest a point and extend Overtime during the duration of their in-vulnerabilities, but neither can capture a point or move a cart in that time.