Simple answer: Yes.
As long as the arrival is programmed in with no discontinuities, valid height and speed constraints and '0' entered into the MCP altitude window for no DH. For an ILS though you would have to arm the APP mode yourself otherwise the aircraft would fly effectively an RNAV VNAV/LNAV approach.
Engage LNAV and VNAV set 0 and watch the fun from the top of drop. I do think you would need to open the speed window after arming the APP mode to control speed but I think that is only for our company policy. I believe the A/C would slow to Vapp itself in APP mode but not add 5 knots. I need to look at that one.
Chances of getting that clearance from ATC? Not much outside of the USA!
