There is special approval required for IAN? Wasn't aware of that, and as far as i know IAN is standard equipment from a certain MSN on.
Anyway, i would argue that LNAV is actually much more precise than raw data, a standard VOR can be up to several degrees off, a DVOR still up to 1°, plus up to 0.4° receiver error added to that, whereas ANP with the 737NG is usually around 0.05. However, yes, coding of the VOR approach in the database has to be correct of course, something which should be taken care of by the database supplier and the quality management system of the airline.
VOR/LOC could be used, however only if there is a co-located DME with the VOR and the VOR is situated below 5000ft, at least according to the FCOM i have, but is really meant for approaches where the VOR is on the arrival end of the runway or well in front of it.