I peeled off the vinyl coated wallpaper on my wall. I am left with a manilla coating. Can I remove this part with a water/vinegar solution?

Yes you can ~ ! Plus scoring it with a sharp knife, without going too deep, helps to get the solution under that paper liner. Wait, spray again, spray again, and then start scraping with a paint spatula. Kids love this job !

Get a "paper tiger" and DIF wallpaper enzyme remover.

Yep, paper tiger is my friend. Zinsser 2966 PaperTiger Scoring Tool for Wallpaper Removal Single Head - Amazon.com (http://www.amazon.com/Zinsser-2966-PaperTiger-Scoring-Wallpaper/dp/B0000DI7WP)

I used fabric softener/water mix in a spray bottle - sprayed it on the wall, waited 5/10 minutes or so, and the wallpaper/adhesive came off in sheets - it was amazing! (room smelled great when I was done too ;) )

Yes.. you can easily remove your wallpaper with water. But if you may not do the same or have any problem. Then you can use other chemicals.