Overlay approaches (VOR/NDB approaches following the same track as a GPS approach) are usually flown in LNAV for horizontal guidance, and either VNAV or V/S for vertical guidance. Autopilot, flight director, and autothrottle use is up to the Captain.
FAA-approved Operational Specifications (OpSpecs) for individual US-certificated airlines may dictate whether or not "pure" GPS approaches may be flown, and whether or not VNAV is approved for use on the final approach segment.