April, the way I read your post, you haven't installed the vinyl yet. If that's correct; good.
Once the nail situation is resolved, the wafer board needs to be covered with yet another wood layer; 1/4" underlayment, which Tom mentions. This is the surface you'll want to install your vinyl on. Vinyl installed directly to OSB or wafer board will "telegraph" all the small variations in the surface of the subflooring to the vinyl.
The underlay needs to be heavily nailed or stapled down, every 4", and the joints should not be at the same points as joints in the subflooring. Once it's attached, use floor patch to skim & fill any gaps between sheets & over nail/staple divits. Anything left unlevel will be visible through the vinyl.
As to the nails popping in the existing material, I'd confront the contractor on this if possible. No matter who does the work, the existing nails should be driven on down below the surface & proper nailing or screws placed nearby each. Good luck