Theoretically, probably yes, but in practice probably too hard.
Reverse osmosis uses quite a lot of power, so that will be the first challenge. Would probably be OK while driving, but otherwise problematic.
The toughest part is likely to be the high levels of fat and detergent in grey water. Some research into the issues associated with RO would be worth while.

I am unsure of why this would be considered. there are few places where water of better quality compared to grey water is not available. Treatment of less contaminated water is simple and inexpensive. We use a combination of chlorination and micro/carbon filtration and that will turn almost any commonly available water into potable water. We do avoid water with very high mineral content and cloudy water, if possible, but even those will suffice, at a pinch.

Cheers,
Peter